Statements About Prime Numbers Flashcards
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?
prime factorization
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
prime
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
integers
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
prime numbers
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.
Fermat’s Last
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
integers
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
unique factorization
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,
prime numbers
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.
prime numbers
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
uniqueness
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
factorization
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.
Carl Gauss
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
prime numbers
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
prime numbers
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
totient
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
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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
Euler’s totient function
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
Gaussian integers
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
Primality
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
prime numbers
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.
Pierre de Fermat
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.
prime numbers
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
p
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
Leonhard Euler
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
prime numbers
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.
prime numbers
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.
prime numbers
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
positive perfect cubes
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
composite numbers
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
Factorial Function
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
Factorial
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
Factorial
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.
Factorial
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
Prime numbers
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.
Factorial
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
Factorial
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?
Factorial
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
Leonhard Euler
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
Perfect numbers
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
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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
powers of two
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
Pierre de Fermat
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
Pierre de Fermat
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.
primality tests
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
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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
Leonhard Euler
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.
Euclid of Alexandria
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
3
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
powers of 2
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
Leonhard Euler
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
Mersenne primes
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) =
even
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
3
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
prime numbers
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
6
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
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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
Triangular numbers
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
3
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
powers of 2
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
3
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
Elements
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.
2
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
2
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
2
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.
triangular numbers
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.
perfect numbers
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
1/2
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.
two