MathDB

Given an integer

a>1

. Let

p_1 < p_2 <...< p_k

be all prime divisors of

a

. For each positive integer

n

we define:

C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k}

A = a^2 + 1

T(n) = A^{C_0(n)} - 1

M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1)

A_n =\frac{T(n)}{M(n)}

Prove that the sequence

A_1, A_2, ...

satisfies the properties: (i) Every number in the sequence is an integer greater than

1

and has only prime divisors of the form

am + 1

. (ii) Any two different numbers in the sequence are coprime.

Problem Statement