# Construction of an infinite set such that any two number from the set are relatively prime

This question is taken from a contest in India.

• Prove that we can construct an infinite set of positive integers of the form $2^{n}-3$, where $n \in \mathbb{N}$, such that any two numbers from the set are relatively prime

I would like to have an answer for this question, and i would also like to know why $2^{n}-3$ has importance here. Can this question be generalized. That is:

• Can we construct an infinite set of positive integers of the form $k^{n} -(k+1)$ such that any two numbers from the set are relatively prime?.
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Given a set of odd primes $\displaystyle S = \{p_1, p_2, \dots, p_m \}$, we can find an $n$ such that $\displaystyle 2^n - 3 = 1 \mod p_i$, by choosing $n = 2 + \prod (p_i-1)$

Thus starting with

$\{2^2 -3,2^3 - 3\}$ we can add a new member which is relatively prime to all the previous members.

I believe a similar argument will hold for other $k$ (when considering $k^n -(k+1)$), since we will have $gcd(p_i,k) =1$.

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First, any common factor of a and b will divide every xn. Furthermore, for any n, $$x_n=(a^{n-1}+a^{n-2}+\cdots+a+1)(a-1)+(b+1).$$ so any common factor of a − 1 and b + 1 will divide every xn. The answer to the question is no if either a, b or a − 1, b + 1 have a common prime factor.