Chapter 7/8 definitions

Question 1

Mersenne Number

Answer 1

Let n∈ℕ, the nth Mersenne number is Mₙ=2ⁿ-1

Question 2

Mersenne Prime

Answer 2

A Mersenne number Mₙ=2ⁿ-1 which is prime

Question 3

The Lucas test

Answer 3

Define a sequence of numbers r₀, r₁, . . . bᵧ
r₀=4 rᵢ₊₁ = rᵢ²-2
Let n be a prime, n ≥ 3. Then Mₙ is prime if an only if rₙ₋₂≡0modn

Question 4

μ =

Answer 4

μ = 1 + √3

Question 5

𝜏 =

Answer 5

𝜏 = 2 + √3

Question 6

order (in an arbitrary ring)

Answer 6

Let R be a ring, n∈{0} and a∈R. The order of a modulo n in R is the smallest natural number d s.t. aᵈ≡1modn. if such d exists. if not a does not have an order modulo n.