Statements About Prime Numbers

Question 1

A form of this procedure is applied to an ideal to determine if it is ramified. This procedure is applied to the bottom argument when defining the Jacobi symbol in terms of the Legendre (“luh-ZHOND”) symbol. Any principal ideal domain is necessarily a domain named for a property of this procedure, called a UFD. This procedure is applied to the input when using the product formula for the Euler totient (“OY-ler TOH-shint”) function. Euclid’s lemma is used to prove a theorem about this procedure. The difficulty of this procedure for large inputs ensures the security of the RSA algorithm. Applying this procedure to two numbers and then checking for shared terms provides a simple way of computing their GCD. For 10 points, the fundamental theorem of arithmetic guarantees the uniqueness of what procedure over the integers?

Answer 1

prime factorization

Question 2

If J is contained in the union of ideals I-sub-one to I-sub-n, and if at most two of those ideals are denoted by this term, then J is a subring of one of the ideals, by a lemma named for the “avoidance” of this property. If the product rs is inside an ideal described by this term, then either r or s must be inside the ideal. A number n has this property if it is the minimum number of times the multiplicative identity of a field can be added to itself to get the additive identity, in which case n is known as the characteristic of the field. Every group whose order has this property is cyclic, since all non-identity elements have order equal to the order of the group, by

Answer 2

prime

Question 3

Rings that contain no zero divisors are “domains” named for these numbers. Any cyclic group of infinite order is isomorphic to these numbers under addition. Polynomial equations that have these numbers as both coefficients and solutions are called Diophantine. Positive examples of these numbers have a unique

Answer 3

integers

Question 4

Only numbers congruent to 1 or 5 mod 6 can have this property. Numbers of this type which are one less than a power of two are named for Mersenne. According to the fundamental theorem of arithmetic, every positive integer can be expressed uniquely as a

Answer 4

prime numbers

Question 5

While using ideal numbers to prove this theorem for regular primes, Ernst Kummer also extended the fundamental theorem of arithmetic to complex numbers. Sophie Germain broke this theorem into two cases and proved it true for her namesake primes. A counterexample to this theorem would imply the existence of non-modular Frey curves, so proving the epsilon conjecture connected this theorem to a special case of the Taniyama–Shimura conjecture. Richard Taylor spent a year helping to correct a flaw in a proof of this theorem given by Andrew Wiles. For 10 points, name this theorem that was written without proof in the margin of a notebook, claiming that x-to-the-n plus y-to-the-n equals z-to-the-n has no solutions when n is greater than two.

Answer 5

Fermat’s Last

Question 6

Only this type of number is allowed as the solution to a Diophantine [dye-oh-FAN-teen] equation. The set of these numbers is not closed under exponentiation due to some results involving negative exponents, and they are the subject of the fundamental theorem of arithmetic. A boldface letter

Answer 6

integers

Question 7

If the ring of integers of Q adjoin tau has this property, then j of tau must be an integer, so Ramanujan’s constant and related numbers are remarkably close to integers. Only finitely many imaginary quadratic fields have this property, according to the Stark-Heegner theorem. Gauss’s lemma on primitive polynomials applies when the polynomials’ coefficients lie in a ring with this property. A Dedekind domain has this property if and only if its class group is trivial. If a ring can be equipped with a Euclidean

Answer 7

unique factorization

Question 8

If both ‘n’ and ‘2n+1’ belong to this class of numbers, then ‘n’ is considered a Sophie Germain type. If ‘n’ is one of these numbers, then any integer raised to the power of ‘n’ is congruent to itself mod ‘n’. By crossing out successive arithmetic sequences of natural numbers,

Answer 8

prime numbers

Question 9

The order of a finite field is always one of these numbers raised to a positive integer. The Green-Tao theorem states that arbitrarily long arithmetic sequences can be found in the sequence containing these numbers. For any integer a and one of these numbers n, a to the n is congruent to a modulo n by Fermat’s little theorem. The Goldbach conjecture claims that any even number greater than two can be expressed as the sum of two of these numbers. According to the fundamental theorem of arithmetic, any integer greater than 1 can be expressed as a product of these numbers. For 10 points, name these numbers whose only divisors are one and themselves.

Answer 9

prime numbers

Question 10

A Torsion-free metric connection must also have this property according to the Fundamental Theorem of Riemannian Geometry. Relations between convex polytopes and polyhedra in Euclidean metric space have this property by Alexandrov’s theorem. Some differential equations whose solutions have this property can be solved by placing a synthetic source and mirroring it across a boundary. This method

Answer 10

uniqueness

Question 11

This operation can be performed using Hensel lifting in the Berlekamp–Zassenhaus algorithm. According to Fisher’s theorem, a statistic is only sufficient if a PDF can have this operation performed on the PDF with the statistic. The householder transformation is the first step on a method to perform the QR type of this operation. A number is called wasteful if the result of this operation has more digits than the number. The general

Answer 11

factorization

Question 12

This man discovered a proof that all positive integers are the sum of at most 3 triangular numbers. He created a formulation of least squares that he used to predict the orbit of Ceres, and he also modified Euclid’s proof of the fundamental theorem of arithmetic. He names an elimination method for solving a system of equations by utilizing matrices along with Wilhelm Jordan. The Central Limit Theorem states that all distributions of repeated events eventually converge to form his namesake distribution, and, as a child, this man supposedly summed the numbers from 1 to 100 in his head. For 10 points, name this German mathematician who is the namesake of the normal distribution.

Answer 12

Carl Gauss

Question 13

If there were a solution to the equation x to the n plus y to the n plus z to the n equals zero, it would be an example of the Wall-Sun-Sun type of these numbers. An elementary proof that the quantity of these numbers less than x divided by x over log x is equal to one was provided by Paul Erdos and Alte Selberg. In 2013, Yitan Zhang found that the number of pairs of this type of number differing by less than 70 million was infinite. By considering all of them multiplied together and then

Answer 13

prime numbers

Question 14

Euler proved that the sum of the reciprocal of these numbers is asymptotic to the log of the log of a limiting n. The quadratic function “x squared plus x plus 41” has these numbers as its range, if the domain is integers less than 40. Euclid explained that the product of a set of numbers with this property, plus one, also has this

Answer 14

prime numbers

Question 15

The limit inferior of the quantity, “this-function-of-n, times log-of-log-of-n, all over n,” yields “e-to-the-negative-gamma.” Kevin Ford proved that for every integer k greater than one, there exists y for which the equation “this-function-of-x equals y” has k solutions. This function of n gives the order of the multiplicative group mod n. Gauss proved that an n-gon is

Answer 15

totient

Question 16

The Riemann zeta function, given Apéry’s theorem, produces an irrational output with this value as the input. A hexagon inside a polygon with this many sides is known as a Lemoine hexagon. The Collatz conjecture multiplies odd numbers by this value, which is also the smallest integer

Answer 16

3

Question 17

Given any integer k, there exists an integer m such that the equation setting this function equal to m has k solutions according to Ford’s theorem. This function is equal to the Möbius transform of the identity function. The number of primitive roots modulo n is equal to this function of this function of n, because the number of generators of a cyclic group is equal to this function on the order of the group. Gauss proved that a

Answer 17

Euler’s totient function

Question 18

Any odd prime equal to one mod four splits over this set, whereas primes congruent to three mod four remain inert in this set. Any number in this set to the power of the norm of a prime nondivisor minus one must be congruent to one mod the prime according to a generalization of Fermat’s little theorem. The biquadratic residue characters of two odd primes in this set are equal to

Answer 18

Gaussian integers

Question 19

A paper by Goldwasser and Kilian produces certificates for entities with this property using elliptic curves. Counterexamples to this property are defined by Baillie and Wagstaff using Lucas sequences and are called “witnesses” or “strong liars” in one context. A paper by Agrawal, Kayal and Saxena titled for entities with “[this property] in P” introduced the

Answer 19

Primality

Question 20

This type of number is the exponent and modulus in Fermat’s Little Theorem. Euclid [yu-klid] proved a method for constructing one of these numbers by multiplying a list of them and adding one. The Sieve of

Answer 20

prime numbers

Question 21

This man’s right triangle theorem states that a rational right triangle cannot have an area that is the square of a rational number. Euler’s totient theorem generalizes the statement “a raised to the p is congruent to a mod p if p is prime,” which is this man’s “little theorem.” This man’s namesake numbers are given by the formula of one plus two to two to the n.

Answer 21

Pierre de Fermat

Question 22

The order of a finite field is always one of these numbers raised to a positive integer. The Green-Tao theorem states that arbitrarily long arithmetic sequences can be found in the sequence containing these numbers. For any integer a and one of these numbers n, a to the n is congruent to a modulo n by Fermat’s little theorem. The Goldbach conjecture claims that any even number greater than two can be expressed as the sum of two of these numbers. According to the fundamental theorem of arithmetic, any integer greater than 1 can be expressed as a product of these numbers. For 10 points, name these numbers whose only divisors are one and themselves.

Answer 22

prime numbers

Question 23

It’s not n, but Wilson’s theorem uses this letter in its representation. One half of a quantity denoted by this letter is multiplied by the inradius to to find the area of a triangle. Fermat’s Little Theorem states that any integer a, taken to a power

Answer 23

p

Question 24

This scientist developed a theory of beams with Daniel Bernoulli, and was the first to use the notation f(x) [f of x]. He solved the Basel problem, which asks for the sum of the reciprocals of the squares. One theorem developed by this man is a generalization to Fermat’s little theorem and involves a raised to the result of his

Answer 24

Leonhard Euler

Question 25

Vinogradov’s theorem established a very large lower bound on a statement about an operation involving these objects, which Harald Helfgott recently claimed to have reduced drastically. Euler proved that the sum of reciprocals of them less than n grows roughly as the log of log of n. Legendre’s conjecture asserts that one of these objects always exists in a certain range. If for all numbers a less than n, a to the power of n minus one equals 1 modulo n, then n is either a

Answer 25

prime numbers

Question 26

A non-zero element of a ring R is one of these elements if it is not a unit, and, for any elements x and y in R, if it divides x times y, it also divides x or y. A number n is one of these numbers if and only if “n minus one factorial” is congruent to “negative one modulo n”, according to Wilson’s theorem. An integer raised to the power of one of these numbers and that integer are congruent modulo the exponent, according to Fermat’s little theorem. When they are equal to “one less than a power of two”, these numbers are named for Mersenne. These numbers can be found using the sieve of Eratosthenes. For 10 points, name these numbers divisible only by themselves and one.

Answer 26

prime numbers

Question 27

This word prefixed by “pseudo” describes n to a base a if a to the n-1 power and 1 are congruent modulo n. Carmichael numbers are examples of those numbers which pass a test described by Fermat’s little theorem. Adding 1 to the product of a finite amount of these was how Euclid proved that there are infinitely many of these numbers. Any positive integer can be uniquely represented as a product of these numbers in a namesake factorization, and 2 is the only even example of these. For 10 points, name these numbers contrasted with the composites, which are only divisible by themselves and 1.

Answer 27

prime numbers

Question 28

If a prime p leaves a remainder of two when divided by three, then Fermat’s Little Theorem can be used to prove that these numbers leave all integer residues modulo p. The third finite difference of this sequence of numbers is always six, and the sum of the first n of these numbers is equal to the nth triangular number squared. Ramanujan pointed out that there were two ways to express 1729 as the sum of two of these numbers, and the smallest case of

Answer 28

positive perfect cubes

Question 29

Carmichael numbers are numbers of this type that satisfy Fermat’s Little Theorem, and the difference between them and their totient is at least two. Arbitrarily long sequences of these numbers can be found by starting at n-factorial plus two up through n-factorial plus n, and there is no jump in the function pi of x at each of them. These numbers are crossed out in the

Answer 29

composite numbers

Question 30

For a prime number p, this operation on p minus one is congruent to negative one mod p according to Wilson’s theorem. The natural log of this function of n is approximately equal to n times the natural log of n minus n. This function can be defined for complex numbers using the gamma function, and it can be approximated by

Answer 30

Factorial Function

Question 31

Wilson’s theorem gives that a number n is prime if and only if this function of quantity n minus one equals negative one mod n. The nth derangement number equals this function of n, divided by e, rounded to the nearest integer. Stirling’s formula approximates this function. The order of the nth symmetric group equals this function of n. The denominator of the nth term of a

Answer 31

Factorial

Question 32

The p-adic valuation of this function of x is equal to the sum over k of (read slowly) the floor of x over p to the k in a formula named for Legendre (“luh-zhon-druh”). This function applied to n minus one is equivalent negative one mod n by Wilson’s theorem. The Poch·hammer symbol denotes the

Answer 32

Factorial

Question 33

When studying hypergeometric functions, one type of this function is represented by the Pochhammer symbol. There is only one function which fulfils the criteria required to sensibly extend this function according to a theorem named after Harald Bohr and Johannes Mollerup. The Kempner function of n gives the smallest number s such that n divides this function of s. Wilson’s theorem states that n is prime if and only if this function of n minus one is equal to minus one modulo n. Stirling’s formula approximates this function for large n. The sum of the reciprocals of this function of n is equal to e. For 10 points, name this function that gives the number of ways of arranging n distinguishable objects, denoted by an exclamation mark.

Answer 33

Factorial

Question 34

A theorem named for the asymptotic distribution of these mathematical objects was independently proved by Hadamard and Vallée-Poussin. Atkin and Bernstein used modulo 60 numbers to develop a construct for identifying these objects up to a certain integer. A certain number is one of these objects, if and only if “n minus one factorial” equals “negative one modulo n”, according to

Answer 34

Prime numbers

Question 35

According to Wilson’s Theorem, this function of a prime-minus-one is congruent to negative one modulo that prime. Subtracting one from this function’s argument results in the differentiable gamma function. The natural log of this function of n is equal to negative n plus n log n for very large n according to Stirling’s approximation. This function is in the denominator of all terms in the Taylor series. This function is used to calculate binomial coefficients with combinations. For 10 points, name this function defined as the product of all positive integers less than or equal to n, which is symbolized by an exclamation point.

Answer 35

Factorial

Question 36

According to Raabe’s formula, the integral, from a to a plus one, of the log of a function that reduces to this function, equals one-half log two pi plus a log a minus a. Wilson’s theorem states that if n is prime, then this function of n minus one is congruent to negative one, modulo n. Stirling’s formula approximates this function as x times ln x minus x, and is used for extremely large values of it. A function that

Answer 36

Factorial

Question 37

The “rising” type of this function is called the Pochhammer symbol. Wilson’s theorem states that this function of a prime minus 1 is one less than a multiple of that prime. The approximation “n ln n minus n” for this function’s natural logarithm is named for Stirling. The coefficients for the “Taylor series of e to the x” are reciprocals of this function, which is generalized by the gamma function and gives the number of permutations of n when performed on n. For 10 points, the product of the first n numbers gives what function of n denoted by an exclamation point?

Answer 37

Factorial

Question 38

This mathematician proved that the Riemann zeta function of s equals the product over all primes p of the reciprocal of one minus p to the negative s in his “product formula.” This mathematician worked 2,000 years after the other namesake of a theorem that establishes a one-to-one correspondence between even perfect numbers and Mersenne primes. This mathematician found that the Riemann zeta function of

Answer 38

Leonhard Euler

Question 39

Most numbers with this property take the form of “1 plus 9 times a triangular number,” and are themselves also triangular. These numbers are the period-1 analogues of sociable and amicable numbers. These values take the form “2 to the quantity ‘p minus 1’, times the quantity ‘2 to the p, minus 1,’” whenever the latter is prime. These numbers are neither “deficient” nor “abundant.” By the

Answer 39

Perfect numbers

Question 40

This number is the largest all-harshad number. This is the maximum number of significant digits in a decimal number that is guaranteed to be fully represented in a 32 bit floating point number. Applied at n equals 2, the equation central to the Euclid–Euler theorem gives this number. In two dimensions, this number is the solution to the kissing problem. The quantity sigma over r all to the this power gives the attractive term in the

Answer 40

6

Question 41

The Ramanujan sum of these numbers is negative one even though the regular sum is divergent. Dyadic rationals are fractions with one of these numbers in the denominator. According to the Euclid-Euler theorem, every even perfect number is the difference of two of these numbers. The number of vertices of a hypercube is one of these numbers. The sum of a row in

Answer 41

powers of two

Question 42

Specific cases of a theorem named for this mathematician were proven by Sophie Germain for her namesake primes. That theorem by this mathematician was proven in its general form using the modularity theorem. Carmichael numbers are counterexamples to this mathematician’s primality test. An equation formulated by this mathematician reduces to the

Answer 42

Pierre de Fermat

Question 43

This mathematician showed that primes of the form, 1 mod 4, can be written as the sum of the squares of two numbers. One of this mathematician’s theorems, extended by Euler, states that for coprime numbers a and b, (pause) a to the power of the Euler totient of b is equal to 1 mod b. This man’s “liars” are also called

Answer 43

Pierre de Fermat

Question 44

An algorithm for this problem that classifies bases as either witnesses or liars is named for Solovay and Strassen. The Baillie–PSW algorithm works because the lists of exceptions to two other criteria have no known overlaps. Pomerance and Lenstra improved the exponent to six in the first polynomial-time algorithm for this problem, which was announced in a 2002 paper by Agrawal, Kayal, and Saxena titled “[this problem] is in P.” False positives in this problem include the Carmichael numbers. A naïve approach to this problem is to divide the input by every positive integer less than it, looking for a remainder of zero. For 10 points, identify this problem of determining whether a given integer has exactly two positive divisors.

Answer 44

primality tests

Question 45

This is the first number that is not an exponent for a Mersenne prime, it is also the third super-prime number. One comedy sketch involves two Scottish men attempting to yell this number into a voice-controlled Lift. In chemistry, this group number contains elements such as copper, silver, and gold. In Minecraft, this number gives its name to an eerie music

Answer 45

11

Question 46

This mathematician proved that the Riemann zeta function of s equals the product over all primes p of the reciprocal of one minus p to the negative s in his “product formula.” This mathematician worked 2,000 years after the other namesake of a theorem that establishes a one-to-one correspondence between even perfect numbers and Mersenne primes. This mathematician found that the Riemann zeta function of

Answer 46

Leonhard Euler

Question 47

The number of steps in a procedure named for this man can be minimized with the method of least absolute remainders. Since their ideals can be generated by an element with a minimal value for their equipped function, this man’s namesake integral domains are always principal ideal domains. Reversing a process named for this man allows one to solve for x and y in Bézout’s (bez-OUTS) identity. This man is the first namesake of a theorem that puts even perfect numbers in correspondence with Mersenne primes, which he co-names with Euler. Consecutive Fibonacci numbers are the worst inputs to this man’s namesake algorithm, which finds the GCD of two numbers. This mathematician gave the first proof of the infinitude of primes. For 10 points, name this mathematician whose number theory proofs appear in his Elements.

Answer 47

Euclid of Alexandria

Question 48

A formula for solving equations of this degree was originated by Niccolo Tartaglia, but is now named for Gerolamo Cardano. Edwards and Patau syndromes are caused when genetic mutations result in this number of a given chromosome, and this number is the first

Answer 48

3

Question 49

Every integer can be expressed as an odd number times one of these numbers. A Mersenne prime is equal to one of these numbers minus one. When these numbers are written in binary, they will always consist of a 1 followed by some number of zeroes

Answer 49

powers of 2

Question 50

This mathematician names a value equal to the integral of the Gaussian curvature with respect to area divided by 2 pi. Along with Euclid, this man names a theorem relating every perfect number and Mersenne prime. For all convex polyhedra, his namesake

Answer 50

Leonhard Euler

Question 51

The product e to the power of the Euler-Mascheroni constant multiplied by the log-base-2 of the log-base-2 of x asymptotically approximates the count of these numbers less than x. That formula is part of the Wagstaff conjecture. The distribution of factors of numbers in this set is described by Gillies’ conjecture. Smooth factors are found by Pollard’s “p minus 1” formula in a project that employs the

Answer 51

Mersenne primes

Question 52

Euclid proved that there is a bijection between the Mersenne primes and the set of perfect numbers that are also this kind of number; it is conjectured, but not known, that all perfect numbers are this kind of number. The Taylor series of cosine contains only terms with this kind of number in the exponent.This term is also used to describe a function f for which f(x) =

Answer 52

even

Question 53

The Möbius-Kantor configuration consists of this many points on each line and this many lines through each point. This is the second Lucas number and the only known prime whose reciprocal has a decimal period of 1. This is the ASCII code for“End of Text” and the retina contains this many types of

Answer 53

3

Question 54

Polignac’s conjecture and Legendre’s conjecture concern gaps between these numbers. For a large N, the probability of randomly generating one of these numbers is about 1 divided by log(N). The Miller-Rabin test and the AKS test are methods of checking if a number belongs to this group, whose

Answer 54

prime numbers

Question 55

There are this many Eisenstein integers with norm 1, and the smallest non-abelian group has this many elements. Pairs of primes that differ by this number are called “sexy,” and the regular polygon with this many sides has nine diagonals. This is the third

Answer 55

6

Question 56

A Turing machine is often called a tuple of this many states, and is also the number of abstraction levels in the OSI model of telecommunications. This is the number of iron atoms in the ideal Prussian Blue pigment. Polygons with this number of sides are the smallest that cannot tessellate. Fully filled f orbitals

Answer 56

7

Question 57

Fang and Chen used Størmer’s theorem on Pell’s equation to negatively answer Sierpiński’s problem asking if 4 of these numbers could form a geometric sequence. An integer n is this type of number if and only if 8 n plus 1 is a perfect square. Ramanujan-Nagell numbers are simultaneously Mersenne primes and these numbers, and setting p equals 1 in Faulbaher’s formula gives these numbers. Gauss’s “Eureka theorem” states that every positive integer can be expressed as the sum of

Answer 57

Triangular numbers

Question 58

Odd numbers are multiplied by this value then added to one in the Collatz conjecture’s computation. This value is the smallest Mersenne prime and the second triangular number. An integer is divisible by this value if the

Answer 58

3

Question 59

In the general statement of Sharkovsky’s theorem, the nth of these numbers is defined to be the “n+1”th smallest number. A positive integer “k” is one of these numbers if applying the bitwise “AND” operator to “k” and “k minus 1” yields zero. For fixed “n”, the sum over “k” of the binomial coefficients “n choose k” is equal to the nth one of these numbers

Answer 59

powers of 2

Question 60

Apéry’s theorem states that the Riemann zeta function at this input is irrational, and Euclid’s postulate of this number describes a circle with any center and radius. Central points in a shape with this many sides define the Euler line, and this integer is the first Mersenne prime

Answer 60

3

Question 61

Pons asinorum was demonstrated in this text before it showed Mersenne primes could be used to find perfect numbers. This text showed that geometric and arithmetic means are not necessarily equal. An algorithm for finding the greatest

Answer 61

Elements

Question 62

This number is the highest integer power found in Pell’s equation. This number raised to the power of its square root gives the transcendental Gelfond-Schneider number. Euler proved that the Riemann zeta function of this number equals pi squared over 6. Adding one to a Mersenne prime will always result in a power of this number. No solution exists for the equation a to the n plus b to the n equals c to the n, when n is greater than this number, according to Fermat’s Last Theorem. This is the largest number whose factorial equals itself, and its square root is equal to around 1.414. For 10 points, name this number that is the base in the binary system.

Answer 62

2

Question 63

This number is the upper bound for the absolute value of iterates in the Mandelbrot set. For a finite set, the cardinality of its power set is equal to this number raised to the power of the cardinality of the original set. This is the Euler characteristic for convex polyhedra, meaning the number of vertices minus the number of edges plus the number of faces is equal to this number. Mersenne primes are one less than a

Answer 63

2

Question 64

Removing a non­leaf vertex from a tree results in this many components, and Mersenne primes are defined as being one less than a power of this number. Differentiating a function of this degree will yield a line. Computer memory is commonly measured in

Answer 64

2

Question 65

The generating function for these numbers is x over quantity one minus x, cubed. Gauss’s “Eureka Theo-rem” is about expressing numbers as sums of three of these numbers. The sum of the first n cubes is the square of one of these numbers. All even perfect numbers are this type of number that can be written as a function of a Mersenne prime. The number of edges in a complete graph is one of these numbers. By looking at a geometric representation of these numbers, one sees that the sum of two consecutive ones is a perfect square. Numbers of this form can be written as n plus 1 choose 2, or n times n plus 1 over 2. For 10 points, iden-tify these numbers like 15, which can be written as 1 + 2 + 3 + 4 + 5.

Answer 65

triangular numbers

Question 66

For any natural number n, there exists only one of these numbers that can be expressed in the form “n-cubed plus 1”. Kanold was the first to show that the amount of these numbers below a given integer n had an asymptotic form of little-O of the square root of n. With the exception of the smallest of these, all known so far can be written as the sum of the cubes of consecutive positive odd integers. For a Mersenne prime with exponent p, a number of this type can be found by multiplying the Mersenne prime by 2 to the power p minus 1, according to the Euler-Euclid conjecture. These numbers are a subset of the triangular numbers, and all numbers of this type found so far are even. For 10 points, name these numbers, such as 496 and 6, that are equal to the sum of their proper divisors.

Answer 66

perfect numbers

Question 67

If p is a Mersenne prime, then p times the quantity “p plus one” times this number yields a perfect number. Evaluating the gamma function at this value yields pi raised to this number, which is the constant of proportionality in the Shoelace Theorem for finding the area of a polygon. This number is conjectured to be the real part of any non-trivial zero of the

Answer 67

1/2

Question 68

A mathematical ring consists of a set and this many operations. This is the smallest positive exponent in the Taylor series expansion of cosine. Mersenne primes are one less than a power of this number. This number is the sum of the infinite series one plus one-half plus one-fourth plus one-eighth and so on. The multivariable function f equals x times y plus one has this degree, as do all quadratic polynomials. For 10 points, identify this prime number, the base of the binary system, which divides all even numbers.

Answer 68

two

Question 69

For a Möbius transformation onto the sphere, there are this many fixed points. For the function at the center of the Collatz Conjecture, inputting the value of four will yield this result. Disjoint neighborhoods exist for any pairs of distinct spaces according to the separability axiom of this number. This is the smallest known Sophie Germain prime. A vertex in the simple, but complete “K sub 3” graph will have this many edges that

Answer 69

2

Question 70

The cyclotomic polynomial of this order is the first eighth-degree cyclotomic polynomial. In a triangle, the Kimberling center named for this number is known as the first isodynamic point. This number is the third non-unity term in the continued fraction representation of pi. A

Answer 70

15

Question 71

The cyclotomic polynomial of this order is the first eighth-degree cyclotomic polynomial. In a triangle, the Kimberling center named for this number is known as the first isodynamic point. This number is the third non-unity term in the continued fraction representation of pi. A

Answer 71

15

Question 72

This mathematician found the only cannonball number, or a number which is both a square-pyramid number and a square number, namely 4,900. He also developed a primality test for Mersenne numbers and holds the record for largest

Answer 72

Edouard Lucas

Question 73

For powers of a prime p, the von Mangoldt function outputs this function of p. The first Chebyshev function is equal to the sum of this function applied to all the primes less than a given number. The limit of this function minus an expression that asymptotically approaches it was conjectured by Legendre (“luh-ZHOND-ruh”) to be slightly greater than one. The integral of one over this function is often offset using a lower bound of two and is denoted Li (“L-I”). According to the earliest form of the prime number theorem, [read slowly] “N over the prime counting function of N” is asymptotic to this function. The difference between this function and the harmonic series approaches the Euler–Mascheroni constant. The derivative of this function is equal to one over x. For 10 points, name this inverse of the exponential function.

Answer 73

Log

Question 74

In one proof of this theorem, the inequality (read slowly) “3 plus 4 cosine phi plus cosine 2 phi is greater than or equal to zero” is used to show that one function has no zeroes on the line “sigma equals 1.” This theorem follows from applying the Wiener-Ikehara theorem to the Dirichlet (deer-eesh-LAY) series for the von Mangoldt function, since it is equivalent to the statement that the Chebyshev function evaluated at x is approximately equal to x. This theorem, which was first proved independently by Hadamard and de la Vallée Poussin, can be improved by using Gauss’s logarithmic integral in place of the function “x over log x.” This theorem gives an asymptotic expression for the counting function pi of x. For 10 points, identify this theorem named for the integers whose distribution it describes, which have exactly two factors.

Answer 74

prime number theorem

Question 75

Examining the spectrum of a quantization of the classical Hamiltonian “H equals xp” yields a result similar to this statement according to the Berry–Keating conjecture. This statement is implied by the stronger Mertens conjecture, which was proven to be false in 1985. Adolf Piltz formulated a “generalized” version of this statement that applies to Dirichlet (“DEE-ree-klet”) L-series, which reduces to the standard version when all terms in the series have numerator 1. This statement, which was introduced in the 1859 paper “On the Number of Primes Less Than a Given Magnitude,” would place a tight bound on the error term in the prime number theorem if it were true. This statement ignores the negative even integers, which are the “trivial zeros.” For 10 points, identify this unproven conjecture that the nontrivial zeros of a certain man’s zeta function all have real part one-half.

Answer 75

Riemann hypothesis

Question 76

The Shannon-Hartley theorem says that the upper bound on information transmission is proportional to this function of a combination of signal and noise. The Prime Number Theorem asserts that the number of primes grows proportionally to n divided by this function of n. It was introduced by John

Answer 76

Log

Question 77

In an algorithm that computes this function with runtime “big-oh of x over the fourth power of log x”, a parameter denoted a is set equal to this function of the fourth root of x so that all kth partial functions with k greater than two vanish. This function violates its usual bound infinitely often, but first does so at a huge number guaranteed to be less than 10-to-the-10-to-the-10-to-964. The Meissel–Lehmer algorithm computes this function that is often replaced with the second Chebyshev function. Ramanujan proved that this function of x minus this function of x over two is at least one in his generalization of

Answer 77

prime counting function

Question 78

A cubic where x squared plus y squared plus z squared equals this number is named after Fermat. As x goes to infinity, the error function tends to this number which is equal to Legendre’s constant for the asymptotic behaviour of the prime counting function. Applying the digamma function to this number gives the Euler-Mascheroni constant. This is the genus of a torus. Cosine squared plus sine squared equals this number which is the number of solutions of any linear equation. For 10 points, name this multiplicative identity, the lowest positive integer.

Answer 78

one

Question 79

One theorem named for this letter states that if there are n physical variables expressed in terms of k quantities, one can create an equation in terms of n minus k different variables that recapitulates the original expression. The nth homotopy group is symbolized by this letter followed by a subscript n. Besides appearing in the name of a theorem named for

Answer 79

pi

Question 80

The first Hardy–Littlewood conjecture asserts that there exists an explicit formula for computing the asymptotic number of certain sets of these things known as “constellations” of these objects. Skewes’s number was used as an upper bound for a condition involving a function named for these things; that function named for these objects is related to the logarithmic integral by a theorem also named for these things. Multiplying finitely many of these things and adding one is a procedure appearing in a proof of the infinitude of these things by Euclid. For 10 points, identify these things, which have specific types named for Germain and Mersenne, and are integers divisible only by one and themselves.

Answer 80

prime numbers

Question 81

Polignac’s conjecture and Legendre’s conjecture concern gaps between these numbers. For a large N, the probability of randomly generating one of these numbers is about 1 divided by log(N). The Miller-Rabin test and the AKS test are methods of checking if a number belongs to this group, whose

Answer 81

prime numbers

Question 82

In 1983, the French firm Matra released a computer of this name in partnership with publisher Hachette noted for its bright red case. A 2014 Supreme Court case often referred to by this common first name ruled that a “do it on the computer” patent for an escrow service was invalid. This is the alphabetically first of a set of characters first described in a 1978 paper co-authored by Ron Rivest. A character of this first name stars in a game developed with the id Tech 3 engine that got a sequel 11 years later subtitled “Madness Returns.” A woman of this name uses a signature when communicating with a man named

Answer 82

Alice

Question 83

Two answers required. The work of these two people prompted three other men to publish a paper in Scientific American offering a $100 prize to anyone who could solve the article’s puzzle. A technique developed by these two people had been already discovered but not published by Clifford Cocks. These two people complained about a system called the S Box that was suspected to have been (+) compromised by the NBS. These two people were also upset that the S Box changed the Lucifer system to rely only on 56 bits. One of these two people suggested that Ralph Merkle also be credited with their key invention, which did not include a

Answer 83

Whitifield Diffie & Martin Hellman

Question 84

The Logjam exploit used knowledge of primes used as the second argument of this operation to precalculate discrete logarithms. Finite fields defined using this operation are used to produce the shared secret in the Diffie-Hellman protocol. The ciphertext is one input into this operation and the product of two large primes is the other in RSA (“R-S-A”) encryption, which applies it after exponentiation. Every ifstatement in every loop of FizzBuzz uses this operation, which is an alternative to binary AND for finding the

Answer 84

modulus

Question 85

The protocol KIMP (kimp) governs systems that manage these things in accordance with OASIS (“oasis”) standards. “Inner” and “outer” examples of these things are derived in HMAC (aitch em ay see), an alternative to digital signatures. One of these things is “expanded” into “round” examples of them when using the Rijndael (RINE-dale) schedule. Only one of these things is used in a “symmetric” form of a process that uses two of them in its “asymmetric” form. These things of length 128, 192, and 256 bits are used in AES (ay ee ess). The discrete logarithm problem allows for the secure transfer of these pieces of information in Diffie-Hellman exchange. In RSA (ar ess ay), these things are generated from two very large prime numbers and are publicly available. For 10 points, give these pieces of information used to encode and decode cryptographic data, similarly to the real-world objects they are named for.

Answer 85

Cryptographic keys

Question 86

One type of these devices named for Kane partially stores information using NMR. These devices may use CNOT to avoid the no-cloning theorem. Pure states of components of these devices can be visualized on a Bloch sphere. These devices suffer from decoherence when they interact with their environment. These devices can search through a list in O of

Answer 86

quantum computer

Question 87

Clifford Christopher Cocks, whose cryptography work paved the way for RSA encryption, earned this award in 1968. Terence Tao became one of the youngest people to win this award in 1987, doing so at the age of 12. In 1992, China became the first team to win its namesake competition

Answer 87

International Math Olympiad Silver Medal

Question 88

One method for this process uses the trapdoor function generated from the discrete logarithm of a point on an elliptic curve. This process typically uses one-way deterministic hash functions. Another form of this process finds the modular inverse of a number coprime to the product of two numbers. The

Answer 88

Encryption

Question 89

One algorithm used to perform this task applies alternating rounds of S and P-boxes allowing for inherent parallelism which isn’t possible when implementing a Feistel network. A textbook named for this process by William Buchanan is one of many to use “Eve” as a third participant. The introduction of the Merkle Damgård construction allowed for the construction of MD5 and SHA-1 (“shah one”), two

Answer 89

Encryption

Question 90

One system in this field involves computing exponentiated elements in cyclic groups and is named for ElGamal whilst another uses elliptic curves to produce more efficient groups for this task. Those systems for this task rely upon an assumption known as the decisional Diffie-Hellman assumption. Many algorithms in this field rely in some form on the hardness of the discrete logarithm problem. One of the first asymmetric systems in this field relies on the fact that prime factorisation is harder than multiplication and is named RSA. A completely secure method in this field is the one-time pad whilst the Caesar cipher is a simple tool in the field. For 10 points what field studies techniques for securely transmitting messages?

Answer 90

cryptography

Question 91

One algorithm for this task uses Wiedemann’s algorithm to row-reduce a sparse matrix over the field with two elements. Another algorithm for this task reduces is it to the period-finding problem, which it then solves using a quantum Fourier transform. One algorithm for this task is based on the birthday paradox and iteratively applies a random polynomial until a value is repeated. Numbers for which this task is easy to compute are called “smooth.” This task, which can be performed using a quadratic sieve or Pollard’s rho algorithm, can be computed efficiently on a quantum computer with Shor’s algorithm. The difficulty of this task provides the security of RSA. Like primality testing, a naive algorithm for this task involves trial division. For 10 points, name this task in which an integer is decomposed into a product of primes.

Answer 91

integer factorization

Question 92

One of these devices is operated using the OpenQASM (“open-kasm”) software package. These devices can accomplish period-finding using the same methods proposed to solve Simon’s problem. These devices often map basis sets using mutually orthogonal matrices with Hadamard gates. Information used by these devices can be represented by a

Answer 92

Quantum computers

Question 93

In RSA encryption, two of these numbers are multiplied to create a public key. These numbers can be identified using the Sieve of Eratosthenes [E-rat-os-tho-neez], and Hillel Furstenberg and Euclid both published proofs demonstrating that there are

Answer 93

Prime numbers

Question 94

A method of testing for this condition named for Solovay and Strassen capitalizes on the efficiency of calculating the Jacobi symbol. An Ulam spiral examines patterns in the occurrence of this property, and numbers with this property can be generated using the Sieve of Atkins or the Sieve of

Answer 94

Primality

Question 95

One algorithm for doing this task generates a sequence using iterated polynomials and investigates the cyclic part of that sequence. Another algorithm for doing this process tries to find smooth perfect squares. Besides Pollard’s rho algorithm and Dixon’s method, another algorithm for doing this process includes a subroutine to find the period of an exponential function using

Answer 95

Prime factorization

Question 96

One type of these objects comprised of anyons [Ann-YON] whose paths make braids in space-time is known as its topological type. Steane codes can be used to overcome a limitation placed on other types of these objects by the no-cloning theorem. These objects make use of

Answer 96

Quantum computers

Question 97

This operation can be performed using Hensel lifting in the Berlekamp–Zassenhaus algorithm. According to Fisher’s theorem, a statistic is only sufficient if a PDF can have this operation performed on the PDF with the statistic. The householder transformation is the first step on a method to perform the QR type of this operation. A number is called wasteful if the result of this operation has more digits than the number. The general

Answer 97

Factorization

Question 98

The 1973 paper that introduced this algorithm discusses Lewis Carroll’s denouncement of an “unfair method” of determining the second-best player of a lawn tennis tournament. This algorithm is only practical for datasets where the number of elements is greater than 4,000,000 since its running time grows on the order of 22n. The simplified recurrence relation for this algorithm is “T of n, is equal to T of n times two tenths plus T of n times seven tenths plus c times n”. Manuel Blum, one of the namesakes of the RSA algorithm, Robert Tarjan, R.W. Floyd et al. introduced this algorithm, which begins by partitioning a list into groups of five. Though this algorithm has a linear runtime, randomized quickselect is often used instead since this algorithm’s overhead is so high. For 10 points, name this algorithm that, as its name indicates, choose the middle element of a list of approximately “middle elements”.

Answer 98

Median of medians

Question 99

Paul Kocher developed a family of methods that perform this task by analyzing timing. Eli Bahim and Adi Shamir developed a set of methods that computes a “differential” between two certain values, which is used in this task. Don Coppersmith created many algorithms for this task, which is done indirectly through “side-channel” methods. This task often relies on solving the

Answer 99

decryption

Question 100

These constructs have arbitrarily long arithmetic progressions according to the Green-Tao theorem. Yitang Zhang used the Bombieri-Vinogradov theorem in a recent paper concerning bounded gaps between these entities. Euler discovered a quadratic polynomial that generates these entities, which are visible on the Ulam spiral. A theorem named for them, which was proved by de la Vallee Poussin and Hadamard, describes their distribution. The difficulty in factoring them is the basis of RSA cryptography, and they can be obtained using the sieve of Eratosthenes. For 10 points, name these numbers that are only divisible by one and themselves.

Answer 100

prime numbers

Question 101

The Rijndael schedule allows for the creation of round ones of these tools. An important problem involving these tools was addressed by the Diffie-Hellman protocol, which allowed for the exchange of these things. PGP uses both symmetric and asymmetric types of these things while AES uses only one of these. These things are stored in the .ssh [“dot s s h”] directory, and PKIs match these to identities using

Answer 101

Cryptographic keys

Question 102

This theorem is extended to the ring of polynomials by the Lagrangian interpolation problem. This theorem is used to discover the secret in both the Mignotte and Asmuth-Bloom schemes for secret sharing. Kurt Godel proved his incomplete-ness theorem by applying Cantor’s pairing function and this theorem, which is also used by the Good-Thomas algorithm to re-index the data output in a fast Fourier transform. This theorem can be used to show that the number of reduced residue classes for some integer m is equal to the Euler totient function applied to m. The RSA algorithm uses this theorem in the pri-vate key to accelerate decryption. An extension of this theorem shows an isomorphism between a quotient ring and a product of quotient rings. This theorem states that for the groups of integers n and a, where n are pairwise coprime integers, there is one integer congruent to a modulo n. For 10 points, name this important theorem from number theory, which was stated around 1500 years ago by the mathematician Sunzi in the country for which it is named.

Answer 102

Chinese remainder theorem

Question 103

The Goldreich-Goldwasser-Micali construction is used in this field to construct a PRF from a PRG. The most widely used algorithm in this field is an implementation of a substitution-permutation network. In this field, Luby and Rackoff showed that the Feistel permutation can be used to transform a PRF into a PRP. A hardcore predicate is used by this field’s Blum-Blum-Shub algorithm, which is used to generate pseudorandom numbers. If integer factorization could be done in polynomial time with Shor’s algorithm, this field’s public-key RSA algorithm, an example of a function that is easy to compute but hard to invert, would be totally screwed. For 10 points, identify this field of computer science concerned with encrypting information.

Answer 103

cryptography

Question 104

The Deutsch-Jozsa algorithm was the first one to use these devices to deterministically solve a problem faster than classical methods, and critical components of them are modeled by Bloch spheres. Using Grover’s algorithm, these devices can search an unsorted database in sub-linear time, and they are difficult to make and maintain due to the phenomenon of

Answer 104

Quantum computers

Question 105

These devices use the Hadamard gate and other unitary matrix operators to manipulate their inputs. The set of problems practically solvable by these devices is BQP, which is thought to be a subclass of P. In 2001, one of these devices was used to show that “15 equals 3 times 5”. To do so, a technique developed by Peter Shor was used, and that technique could one day allow these devices to break RSA. D-Wave has built one type of these devices, which performs annealing using 128 qubits (“Q-bits”). For 10 points, name these devices, first proposed by Richard Feynman, that use distinctly non-classical phenomena to execute algorithms.

Answer 105

Quantum computers

Question 106

A counter mode named after this mathematician is under development as a variant of one named after Galois. One measurement of curvature named after this mathematician is an average of minimum and maximum curvatures. This mathematician proved specific cases of Fermat’s Last Theorem for primes under 100, and their tutelage under Lagrange led them to work on vibrations of (+) plane laminae. Numbers named after this mathematician are often paired with so-called “safe primes” and can be generated by p-1 all over 2, where p is a safe prime. For 10 points, identify this mathematician who died from breast cancer and whose namesake identity shows a factorization of a sum of

Answer 106

Sophie Germain

Question 107

This mathematician’s unpublished manuscripts were confiscated from Guglielmo Libri’s library after Libri fled Paris to escape arrest for habitual theft from public libraries. This mathematician was the first to give the correct differential equation to describe Chladni [KLAHD-nee] figures. This mathematician’s namesake numbers correspond to “safe primes” useful in cryptography. In a footnote, Legendre credited this mathematician with the theorem that, if x, y, and z are solutions to “x-to-the-p plus y-to-the-p equals

Answer 107

Sophie Germain

Question 108

If p were a program that performs this task, then one could construct a program q such that p, evaluated on q with q’s input being the source code of q, outputs a contradiction; hence, this task is impossible in general. What is this problem concerning Turing machines, which, if it were decidable, would imply the solution of Goldbach’s conjecture and the computability of the Busy Beaver function?

Answer 108

solving the halting problem

Question 109

This painting is on the cover of the novel Uncle Petros and Goldbach’s Conjecture. On the right side of this painting, one can read a shop window that says “Art Co.” Unusually, pinholes can be seen around the central objects of this painting, probably due to the artist’s use of a pinboard. Two street lamps lie in the upper corners of this painting, in which black lines depict a heavy rain. This painting’s title was taken from

Answer 109

I Saw the Figure 5 in Gold

Question 110

Symplectic manifolds have dimensions with this property. Integers described by this term are not taken to the namesake value in the “3n+1” Collatz conjecture. The sum of degrees of a graph’s vertices has this property, by the handshake lemma. The alternating group consists of all permutations described by this term, which have positive sign. All of these numbers can be written as a sum of two primes, according to Goldbach’s conjecture. It is an open question whether or not all perfect numbers have this property. Functions described by this term include cosine and x-squared, and are symmetric about the y-axis. For 10 points, what term describes numbers divisible by 2?

Answer 110

even numbers

Question 111

Removing these numbers from Pascal’s triangle results in the formation of Sierpinski’s triangle. This property is possessed by the degrees of all the vertices in an Eulerian graph. It is hypothesized that all perfect numbers also satisfy this condition, and these types of numbers can be expressed as the sum of two

Answer 111

Even

Question 112

A function that has this property on the non-negative integers must have this property everywhere when the conditions for Carlson’s (carl-SAWNS) theorem are met. Schwarz’s lemma describes the rigidity of certain functions that have this property at the origin. The minimum modulus principle applies to a holomorphic function f on a domain U, only if f has this property nowhere on U. The Riemann zeta function has this property

Answer 112

vanishing

Question 113

Any sequence of values tending to infinity can be characterized as a set of these points in the Weierstrass (“VYE-er-strahss”) factorization theorem. A function and its sum with a function that it dominates have the same number of these points within a contour by Rouché’s (“roo-SHAY’s”) theorem. The poles of a meromorphic function f are these points of the function one over f. The negative even numbers are “trivial” examples of these points for a function whose others are known to lie in the “critical strip.” The real parts of these points for the zeta function are all conjectured to equal one-half by the Riemann hypothesis. Any nonconstant polynomial has one of these points by the fundamental theorem of algebra. For 10 points, name these points that, for a real single-variable function, represent intersections with the x-axis.

Answer 113

zeros

Question 114

In probability theory, this letter names a discrete distribution whose pdf equals “1 over x to the shape parameter” times the sum from “i equals 1 to n” of “1 over i to the shape parameter”. Observations of Shafranov kink instabilities led Lyman Spitzer to target a fusion reactor known by this Greek letter to retract data in the 1960s. A quantity named for this letter is measured at the slipping plane of an interfacial

Answer 114

zeta

Question 115

Noether’s generalization of a result named for this mathematician says that for a finite Galois extension L over K, the first Galois cohomology of L-times is trivial. This mathematician proved an order-reversing bijection between radical ideals and algebraic sets. Another theorem proved by this mathematician states that the polynomial ring over a

Answer 115

David Hilbert

Question 116

According to the Birch and Swinnerton-Dyer conjecture, this number will be the rank of an elliptic curve whose L-function has a simple pole. Although they are commonly denoted by a (emphasize) different number, this is the (emphasize) number of elements in both the first and last entries in a short exact sequence. This number is the class number of any principal ideal domain. The Riemann hypothesis may be proven with the use of a hypothetical structure called the “field” with this many elements. This is the size of the largest conjugacy class of any elliptic curve group, as well as any abelian group. The fundamental group of any simply-connected space has this many elements. A simple group has this many proper normal subgroups. The smallest possible group has, for 10 points, what number of elements, which is also the number of identity elements in any group?

Answer 116

one

Question 117

The Fourier transform is an automorphism on a space of these things named for Schwartz. The uniform boundedness principle and the Hahn–Banach (BAH-nuck) theorem are central to a branch of analysis named for these things, which is concerned with topological vector spaces. Any one of these things that is both entire and bounded must be constant, and these things are called entire if they are holomorphic over the whole complex plane. Weierstrass (VYE-ur-SHTROSS) names one of these things which serves as a common counterexample in analysis. The Riemann (REE-mon) zeta one of these is the subject of the Riemann hypothesis. For 10 points, identify these things which can be bijective, surjective, and injective, and which can be composed with each other.

Answer 117

function

Question 118

It’s not trivial, but the module generated by the empty set is referred to by this word, and is the identity of modules under direct product. In a field, the object referred to by this word is the only one that does not have a multiplicative inverse. This word describes the image of the kernel of a linear transformation. The Riemann hypothesis is about the nature of numbers which output this value when input into the zeta function. When this number is taken as a polynomial, it has degree negative infinity, though other constants considered as polynomials have degree equal to this number. The y-intercept of a function is its value at this x-coordinate. For 10 points, the roots of a polynomial are where it takes on what value?

Answer 118

zero

Question 119

A tensor product of two objects named for this man is isometrically isomorphic to a space of operators named for him and Schmidt. Wrightman’s axioms suggest that a construct named for this man exists that is acted upon by the Poincaré group in a unitary manner. A result named for this man states that a polynomial ring over a Noetherian ring is itself Noetherian. A finite- or-infinite dimensional space in which an inner product is defined is named for this man. A problem in this man’s program was shown to be impossible by Godel’s incompleteness theorem, and he included the continuum hypothesis and the Riemann hypothesis in another outline of work for the next century. For 10 points, name this German mathematician who laid out a list of 23 difficult-to-solve problems at the beginning of the 20th century.

Answer 119

David Hilbert

Question 120

Physicists use this mathematical object to regularize divergent sums, setting its value for argument 1-k to be minus the kth Bernoulli number over k. By Voronin’s theorem, this function exhibits universality, so that it can approximate any non-zero holomorphic function arbitrarily closely inside its critical strip. A generalization of one conjecture about this function is stated in terms of Dirichlet L-functions. For a complex number s with real part greater than 1, this function equals the sum of “n to the negative s” taken over all positive integers n. It is an important open question whether all of its nontrivial zeroes have real part one-half. For 10 points, name this function that is the subject of the Riemann hypothesis.

Answer 120

Riemann zeta function

Question 121

A theorem of Kolmogorov named for three of these constructs can be used to prove the strong law of large numbers. The Borel–Cantelli lemma gives a condition on one of these things consisting of measures of sets that implies their lim sup (“soup”) has measure zero. The Weierstrass M-test can be used to show that one of these things defines a function. For real-valued examples of these constructs, unconditional

Answer 121

series

Question 122

Local functions named for this letter defined for finite field curves are generalized by the Weil (“vay”) conjectures. The norms of algebraic number fields define functions named for this letter and Richard Dedekind (“DAY-duh-kind”). Euler’s (“OY-lur’s”) solution to the Basel (“BAH-zel”) problem inspired one mathematician to define a function named for this letter, which is equal to “pi squared over six” at two. Evaluating a function named for this letter at “s equals 1” produces the harmonic series. The roots of a function named for this letter would provide a strong asymptotic bound on the distribution of prime numbers. It is conjectured that the zeros of a function named for this letter are either negative even integers or have real part one half. For 10 points, name this Greek letter that denotes a function named for Bernhard Riemann.

Answer 122

zeta

Question 123

Euler’s (“OIL-er’s”) reflection formula is a simple way to find the value of the gamma function at this number; that value is the square root of pi. Weierstrass’s (“VIE-er-strass’s”) variant of trig sub simplifies integrals using a variable equal to the tangent of this number times theta, which equals sine theta over the quantity “one plus cosine theta”. This number is the real part of all known non-trivial zeros of the

Answer 123

1/2

Question 124

This operation is applied to a function f on the boundary of the unit disk in the Schwarz integral, which computes f on the whole unit disk. The output of this operation on a function is set equal to a Cauchy (“COH-shee”) principal value in the [emphasize] first Kramers–Kronig relation. The Laplace (“luh-PLASS”) transform of the unit step function converges if its argument lies in the half-plane on which this operation is positive. The last step of phasor (“phaser”) analysis is to apply this operation to the phasor of interest to get a physical answer. If this continuous but non-analytic operation is applied to a complex number z, the result is [read slowly] “z plus z-bar, all over two.” Every known nontrivial zero of the Riemann zeta function returns one-half under this operation, which yields “cosine of x” when applied to “e to the ix.” For 10 points, name this operation that returns the non-imaginary part of a complex number.

Answer 124

taking the real part of a number

Question 125

Three of Andre Weil’s (“vay’s”) conjectures about generating functions named for this letter were solved by Alexander Grothendieck (“GRO-tun-deek”), who failed to show that the magnitude of alpha was bounded. Though not pi, a function named for this letter determines the terms in a formula that provides an explicit means of calculating the number of primes less than x. When the input is one, the best-known function named for this letter reduces to the

Answer 125

zeta

Question 126

This man names a theorem whose proof often uses the result that certain locally uniformly bounded families of functions are normal, called Montel’s theorem. The Möbius transformations are the automorphisms of an object named for this man, on which any rational function can be extended to a continuous function. This man is the alphabetically-later namesake of the equations “u-sub-x equals v-sub-y” and “u-sub-y equals negative v-sub-x” that give necessary conditions for a complex function to be holomorphic. This man names a function given by the infinite series “one over n to the s,” whose “critical strip” contains all of its nontrivial zeros. For 10 points, a Millennium Prize Problem concerns what German’s namesake zeta function?

Answer 126

Bernhard Riemann

Question 127

Multiples of numbers in this set are used to create a Beatty sequence. Apery proved that the Riemann zeta function at three is this type of number. Dedekind cuts act on this set’s complement to create this set of numbers, which when inputted into Thomae’s function returns a value of 0. Hippasus was supposedly drowned for discovering this

Answer 127

irrational numbers

Question 128

On the graph x^x, this function of e is the x value of the absolute minimum, while this function of the golden ratio is one less than itself. This non arithmetic function must be satisfied by every element in a set in order for it to be defined as a field. Taking the Riemann Zeta function of 1 is equal to the infinite sum of this function of the natural numbers, which is equal to

Answer 128

reciprocal

Question 129

The Riemann zeta function is equivalent to this mathematician’s namesake product indexed by prime numbers. This mathematician names a method of numerical integration that is the simplest Runge-Kutta method. The limiting difference between the harmonic series and the

Answer 129

Leonhard Euler

Question 130

The Riemann zeta function of this number is equal to Apéry’s constant. In every dimension greater than four, there are only this many regular polytopes. This is the smallest number in the only prime triplet. Polygons with this many sides are the subject of

Answer 130

3

Question 131

In probability theory, this letter names a discrete distribution whose pdf equals “1 over x to the shape parameter” times the sum from “i equals 1 to n” of “1 over i to the shape parameter”. Observations of Shafranov kink instabilities led Lyman Spitzer to target a fusion reactor known by this Greek letter to retract data in the 1960s. A quantity named for this letter is measured at the slipping plane of an interfacial

Answer 131

zeta

Question 132

This mathematician’s only paper on number theory derives an exact formula for the prime-counting function. One-dimensional complex manifolds are collectively named for this mathematician. The late Michael Atiyah announced a nonsensical proof of a conjecture by this mathematician. That conjecture posits that the line with real part one-half contains every non-trivial zero of this mathematician’s

Answer 132

Bernhard Riemann

Question 133

If b is one of these numbers, then “a raised to the b power” is transcendental as long as a is not zero or one according to Gelfond’s theorem. Apery’s constant, equal to the Riemann zeta function of three or the sum of n from one to infinity of one over n cubed, is one of these numbers. The set of these numbers is often indicated with a blackboard

Answer 133

irrational numbers

Question 134

In a 1977 paper, Stephen Hawking calculated the log of the partition function to be a linear combination of one of these functions and its derivative; that is a form of regularization named for these functions. For an operator A, one of these functions is given by the Mellin transform of the kernel of the heat equation “d F d t plus A F equals zero.” Because the Bose-Einstein distribution resembles the integrand in a representation for one of these functions, that one of these functions appears in the formula for the

Answer 134

zeta functions

Question 135

This person names a line that passes through the orthocenter, centroid, and circumcenter of a non equilateral triangle. This mathematician’s equations deal with liquid flow with no viscosity, and he connected prime numbers to the Riemann zeta function. Graph theory was introduced by this man in the Seven Bridges of Königsberg problem, which he proved to be impossible. For 10 points, name this Swiss mathematician whose namesake number is equal to about 2.718. (GP)

Answer 135

Leonhard Euler

Question 136

A 1956 work of Martin Gardner is titled for this academic subject, “Magic, and Mystery”. One measure of collaboration among people in this field is the Erdős [AIR-dish] number. Honors given for work in this subject include the Émile Picard Medal, the Abel Prize, and the

Answer 136

mathematics

Question 137

For inputs with real part between ½ and 1, Voronin’s Universality Theorem shows that, under certain technical conditions, this function is able to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The probability that k random numbers have a greatest common divisor of 1 is equal to the reciprocal of this function of k. Euler solved the

Answer 137

Leonhard Euler

Question 138

In these systems, “dark” or “bright” solitons can form depending on whether the interaction is attractive or repulsive. For certain scattering processes in these substances, a bound state known as a Feshbach resonance can occur. The critical temperature for one of these substances to form is inversely proportional to the Riemann zeta function of three-halves to the two-thirds power. The superfluidity of helium-4 is related to the formation of one of these substances. The first step in creating these substances often involves placing atoms in a magneto-optical trap. The first example of these substances was observed by laser cooling a vapor of rubidium atoms. For 10 points, name this substance, a gas of integer-spin particles at a temperature near absolute zero.

Answer 138

Bose-Einstein condensate

Question 139

Apéry’s theorem states that the Riemann zeta function at this input is irrational, and Euclid’s postulate of this number describes a circle with any center and radius. Central points in a shape with this many sides define the Euler line, and this integer is the first Mersenne prime.

Answer 139

3

Question 140

This technique is used on algebraic approximants to build solutions to power flow equations in the HELM method. This process can be performed starting from a given point and proceeding along rays, a configuration known as the Mittag-Leffler star. It cannot be performed on power series which contain gaps between their nonzero terms according to the Hadamard-Ostrowski theorem; it also cannot be performed if all points on the circle of convergence for a function are singular. Doing this process along a curve with overlapping disks is possible via the monodromy theorem. Performing this technique on the regularized Riemann zeta function is done to show that the sum of the natural numbers converges to -1/12. The gamma function can be defined over the entire complex plane by using, for 10 points, what technique that extends the domain of a complex-differentiable function?

Answer 140

analytic continuation

Question 141

This number is the highest integer power found in Pell’s equation. This number raised to the power of its square root gives the transcendental Gelfond-Schneider number. Euler proved that the Riemann zeta function of this number equals pi squared over 6. Adding one to a Mersenne prime will always result in a power of this number. No solution exists for the equation a to the n plus b to the n equals c to the n, when n is greater than this number, according to Fermat’s Last Theorem. This is the largest number whose factorial equals itself, and its square root is equal to around 1.414. For 10 points, name this number that is the base in the binary system.

Answer 141

2

Question 142

The number density of particles in these systems contains a constant equal to 16 pi times the Riemann zeta function of 3. In an accelerating reference frame, vacuums resemble one of these systems with a temperature proportional to the acceleration. One equation describing these systems is derived by counting the number of standing waves per unit volume, then calculating the average energy per mode with a Boltzmann distribution. That law was derived because using the equipartition theorem to describe these systems leads to them having

Answer 142

blackbody

Question 143

The n+1th [N-plus-oneth] member of a series of numbers named for a scientist with this surname appears over n plus one in a formula for calculating the Riemann Zeta function of minus n. The probability distribution of a random variable with success probability p and failure probability one minus p is named for a scientist with this surname. The binomial distribution can be considered as a number of that scientist’s

Answer 143

Bernoulli

Question 144

The stick-breaking process is a constructive algorithm used to generate samples from a stochastic process named after this mathematician. This mathematician also names a multivariate continuous probability distribution defined on a n-dimensional simplex, which is the conjugate prior of the multinomial distribution. In analytic number theory, the Riemann zeta function is a simple non-trivial example of his type of series, which are sums over n of a-sub-n over n to a complex power s. This mathematician also names the eta function, a simple example of one of his L-functions. In contrast to Neumann boundary conditions, boundary conditions named after this man specify a function’s value on the boundary of its domain. For 10 points, name this man who codified, and sometimes names, the pigeonhole principle.

Answer 144

Peter Gustav Lejeune Dirichlet

Question 145

This number is the value of the Riemann zeta function which yields Apéry’s constant and Waring’s problem asks whether every positive integer can be written as the sum of nine or fewer numbers raised to this power. Stewart’s theorem relates the side lengths of a polygon with this number of sides to its cevian. It is the only prime triangular number and the only prime one less than a perfect square. This number is the minimum number of points needed to define a circle and the sum of interior angles of a polygon with this number of sides is 180 degrees. For 10 points, name this integer, the first odd prime number.

Answer 145

3

Question 146

Numerical computations of this function often make use of the Lanczos approximation. Holder’s theorem states that this function does not satisfy any algebraic differential equation with rational function coefficients. Its characterization as the unique meromorphic function which is log-convex and obeys certain functional equations is due to Bohr and Mollerup. It’s not sine, but this function appears in the functional equation for the Riemann zeta function. For real arguments, it can be defined using the Euler integral of the second kind. This function has poles at all

Answer 146

gamma function

Question 147

If p is a Mersenne prime, then p times the quantity “p plus one” times this number yields a perfect number. Evaluating the gamma function at this value yields pi raised to this number, which is the constant of proportionality in the Shoelace Theorem for finding the area of a polygon. This number is conjectured to be the real part of any non-trivial zero of the

Answer 147

one-half

Question 148

Viete’s formula can be used to compute this using an infinite product, and the Euler-Poisson integral is equal to the square root of this. The Riemann zeta function evaluated at s=2 is equal to this quantity squared over six. Barbier’s theorem states that every curve of constant width has a perimeter this many times its width. The Chudnovsky algorithm was used in calculations of this to 10 trillion digits. The Buffon needle is notable for being able to approximate this, and e to the power of i times this number is negative 1.Name this transcendental number, often approximated 3.14, which is used in finding the area of a circle.

Answer 148

pi

Question 149

The particle density divided by the Riemann zeta function all raised to the two-thirds power is proportional to the transition temperature of these systems. The nonlinear equation modeling their behavior has the form of the Schrödinger equation with the addition of an interaction term and is known as the Gross-Pitaevskii equation.A type of this substance in which the particles are confined to one dimension is named after Tonks and Girardeau, and this supercooled state of matter was first synthesized by Cornell and Wieman in rubidium atoms. For 10 points, identify this doubly-eponymous “fifth state of matter.”

Answer 149

Bose-Einstein Condensate

Question 150

The maximum possible volume for the geometric figure known as Gabriel’s Horn is this quantity. The square root of two divided by two is the first product of an infinite sequence of terms named for Viete [vee-ET] that expresses this quantity. The value of the Riemann zeta function at two is equal to this quantity squared divided by six. The period of the tangent function is equal to this quantity. The number e raised to the product of i and this quantity is equal to negative one. For 10 points, name this transcendental number, the ratio of a circle’s circumference to its diameter.

Answer 150

pi

Question 151

This man names a theorem whose proof often uses the result that certain locally uniformly bounded families of functions are normal, called Montel’s theorem. The Möbius transformations are the automorphisms of an object named for this man, on which any rational function can be extended to a continuous function. This man is the alphabetically-later namesake of the equations “u-sub-x equals v-sub-y” and “u-sub-y equals negative v-sub-x” that give necessary conditions for a complex function to be holomorphic. This man names a function given by the infinite series “one over n to the s,” whose “critical strip” contains all of its nontrivial zeros. For 10 points, a Millennium Prize Problem concerns what German’s namesake zeta function?

Answer 151

Bernhard Riemann

Question 152

Any sequence of values tending to infinity can be characterized as a set of these points in the Weierstrass (“VYE-er-strahss”) factorization theorem. A function and its sum with a function that it dominates have the same number of these points within a contour by Rouché’s (“roo-SHAY’s”) theorem. The poles of a meromorphic function f are these points of the function one over f. The negative even numbers are “trivial” examples of these points for a function whose others are known to lie in the “critical strip.” The real parts of these points for the zeta function are all conjectured to equal one-half by the Riemann hypothesis. Any nonconstant polynomial has one of these points by the fundamental theorem of algebra. For 10 points, name these points that, for a real single-variable function, represent intersections with the x-axis.

Answer 152

zeros

Question 153

This man extended Gauss’s ideas about curvature in three-dimensional space to surfaces in higher dimensional space. In an 1859 paper, he used a complex variable to analyze a function whose zeroes he posited are on the critical line, meaning they have a real part of one-half. He names a method in which points are taken on a function to create rectangles, which are added together to approximate area. For 10 points, what German man who names a zeta function has “sums” that can evaluate a definite integral?

Answer 153

Bernhard Riemann

Question 154

A form of this operation named after Jean-Gaston Darboux was extended to higher dimensions by Thomas Stieltjes (STEEL-jee-iss). An inequality for switching this operation with the limit inferior was proven by Pierre Fatou. Linear combinations of indicator functions called simple functions are used to define a form of this operation for which the dominated and monotone convergence theorems hold. This operation is the same for two functions that differ only on a null set. One form of this operation is defined by using the Lebesgue (luh-BEG) measure. In some cases, Fubini’s theorem allows for two instances of this operation to be interchanged. A form of this operation is defined when upper and lower sums named for Bernhard Riemann agree. For 10 points, name this operation from calculus that finds the area under a curve.

Answer 154

integration

Question 155

This mathematician’s paper “On the Hypotheses Which Lie at the Bases of Geometry” outlined his version of differential geometry. This student of Carl Friedrich Gauss proposed that every nontrivial solution to one problem lies on a line in the complex plane with real part one half. That namesake

Answer 155

Bernhard Riemann

Question 156

This man showed that there exists a biholomorphic function onto the open unit disk in his mapping theorem. This man’s lecture “On the Hypotheses on which Geometry is Based” outlined his namesake field of geometry. This mathematician names a set of equations with Cauchy that determine the holomorphism of a complex function, and his namesake

Answer 156

Bernhard Riemann

Question 157

One construct named for this man represents an extension of the complex numbers with a value for infinity, which can permit division by zero in some cases. With Augustin Cauchy, this man names a pair of partial differential equations that can be used to determine if a complex function is holomorphic. In this man’s only paper on number theory, he describes a function whose nontrivial

Answer 157

Bernhard Riemann

Question 158

This mathematician wrote the textbook “Cours d’analyse” [KOOR dah-nah-LEASE] which marked one of the first steps in the evolution of calculus since the mid-eighteenth century. He almost single-handedly pioneered complex function theory and he lends his name to a

Answer 158

Augustin-Louis Cauchy

Question 159

A function named for this mathematician and symbolized by a Greek letter generalizes a simpler function that sums reciprocals of squares from one to a given positive integer. A conjecture named for him states that all roots of the former function occur at negative even integers or at arguments with real part one-half, this mathematician’s namesake

Answer 159

Bernhard Riemann

Question 160

If a class of objects named after this mathematician are simply connected, then they must be isomorphic to either another object named for this man, the complex plane, or the unit disk. Those objects are also the subject of a theorem he co-names with Roch. A problem named for this mathematician can be generalized to involve all Dirichlet L-functions. His namesake

Answer 160

Bernhard Riemann

Question 161

A “combinatorial” version of a theorem named for this man was first proved by James Cannon, who did so by decomposing actions on a certain object to actions that have undergone Mobius transformations. That theorem named for this mathematician can be extended to both one-sided and two-sided boundary arcs by making use of the Schwartz reflection principle, and is used to build analytic continuations. That theorem named for this mathematician was generalized to the “uniformization theorem” by Felix Klein, and states that there is a bi·holo·morphic mapping from a non-empty simply connected open subset of the complex plane to the open unit disk. A pair of differential equations named for this man and Cauchy gives necessary and sufficient conditions for functions to be complex differentiable. For 10 points, identify this German mathematician who names a “zeta” function.

Answer 161

Bernhard Riemann

Question 162

Along with Liouville [“loo-ee-vill”], this mathematician names an integral in fractional calculus that associates each function f with a set of iterated antiderivatives of f. This man’s mapping theorem assures the existence of a biholomorphic map from each simply-connected open subset of the complex plane to the open unit disk. His namesake

Answer 162

Bernhard Riemann

Question 163

The “exotic” variety of these entities are homeomorphic, but not diffeomorphic, to their standard n-dimensional Euclidean analogues. The complex plane plus infinity can be represented on one of these entities named for Bernhard Riemann. The Kepler conjecture concerns the maximal packing of these solids, and points in 3-space can be described using a set of coordinates, which includes two angular values, that are named for these solids. The volume of one of them is equal to four thirds pi r squared. For 10 points, name these solids whose cross sections are circles.

Answer 163

spheres

Question 164

This man extended Gauss’s ideas about curvature in three-dimensional space to surfaces in higher dimensional space. In an 1859 paper, he used a complex variable to analyze a function whose zeroes he posited are on the critical line, meaning they have a real part of one-half. He names a method in which points are taken on a function to create rectangles, which are added together to approximate area. For 10 points, what German man who names a zeta function has “sums” that can evaluate a definite integral?

Answer 164

Bernhard Riemann

Question 165

These equations are discretized by generating cells with pressure at the center and velocity components on the edges in the Marker and Cell method. Through the use of a bound on the L4-norm, Ladyzhenskaya (“lah-DEE-zhen-skaya”) showed that these equations admit global unique solutions in two dimensions. The fact that quantities under a version of this equation evolve either supercritically or non-coercively was used by Terence Tao to show that there exists a solution with finite-time

Answer 165

Navier-Stokes equations

Question 166

A metric named for this mathematician, which is the subject of the Schwarz–Pick theorem, has constant curvature of negative one on a disk used to model hyperbolic geometry also named for this man. A measure-preserving transformation of a finite measure space will cause almost every point to return arbitrarily close to itself by a theorem named for this man. On contractible domains, a closed differential form is exact by his namesake lemma. A statement named for this mathematician was proven as a corollary of Thurston’s geometrization conjecture using Ricci flows and remains the only solved Millennium Prize Problem. For 10 points, name this French mathematician whose conjecture about manifolds homeomorphic to the 3-sphere was proven by Grigori Perelman.

Answer 166

Henri Poincare

Question 167

This is the last numerical digit of Graham’s number, and it is the lowest number of dimensions of a known exotic sphere. While a plane can be colored with four colors, this is the smallest number needed to color a torus. The cross product in this many dimensions does not satisfy the Jacobi identity, and the cross product is only

Answer 167

7

Question 168

A key step in the proof of this theorem was proving the “no local collapsing theorem”, which eliminated some of the difficult asymptotic behaviour presented by the cigar-type soliton. This result can be proved in higher dimensions by using the “Whitney Trick” to prove the h-cobordism theorem. This result, a special case of

Answer 168

Poincare

Question 169

This man is the alphabetically prior namesake of a number theoretic technique for bounding sums by writing them as integrals over the unit circle and then distinguishing between major and minor arcs. Applying that technique named for this man to the von Mangoldt function gives a proof of Vinogradov’s theorem. This man is the alphabetically prior namesake of an operator that takes a function and returns the maximum average value of that function on balls centered at a given point. That operator named for this man is used to prove the

Answer 169

G. H. Hardy

Question 170

Michael Kwok developed a program to assist in solving this problem that relies on NewPGen. Advancements on this problem were made by the GPY paper, but mathematicians were unable to extend GPY to achieve a 4/7 ratio developed in the 1980s by Bombieri et al. Terence Tao led an effort to optimize a lower bound in a partial proof of this statement. A constant value of about 0.66 that shares its name with this statement was developed by Hardy and Littlewood; another generalization of this statement was made in the 19th century by Alphonse de

Answer 170

twin primes conjecture

Question 171

A result involving the lengths of constructs with this property was proven using Goldston, Pintz, and Yildirim’s result on small prime gaps to show that a “transference principle” is applicable. Erdős (“AIR-dich”) conjectured that if the sum of reciprocals of a sequence diverges, that sequence must contain arbitrarily large subsequences with this property; that conjecture would generalize Szemerédi’s (“seh-meh-RAY-dee’s”) theorem. An observation regarding the distribution of primes in two sequences with this property is called

Answer 171

arithmetic

Question 172

Viggo Brun used one of these methods to prove the existence of his namesake constant. The Bombieri-Vinogradov theorem, which estimates the error in Dirichlet’s theorem, was proven using a “large” one of these methods. The size of a set is expressed as a sum involving the Möbius function in the Legendre identity, which is the basis for many of these methods such as one named for Atle Selberg. Yitang Zhang used these methods to prove the bounded gaps conjecture. These methods are called “combinatorial” when they are based on the inclusion-exclusion principle. Modern methods named for Sundaram and Atkin improve upon an ancient one of these methods in which multiples of small integers are successively removed from a sample set until only the prime numbers remain. For 10 points, name these methods of constructing specific subsets of the natural numbers, one of which is named for Eratosthenes.

Answer 172

sieves