Let $(a,b)$ be a pair of coprime positive integers with $a$ being even. Are these conditions sufficient to prove that there exist infinitely many positive integers $n,$ such that $(a^n+1,b^n+1)=1$ ?

## 1 Answer

I think that this sort of question was originally asked by Ailon and Rudnick, but they use $-1$ instead of $+1$ and asked if $\gcd(2^n-1,3^n-1)=1$ for infinitely many $n$. In this setting, for more general $a$ and $b$, the right question/conjecture would be $\gcd(a^n-1,b^n-1)=\gcd(a-1,b-1)$. They prove something stronger if you replace $\mathbb Z$ with $\mathbb C[t]$. Here's the Ailon-Rudnick paper:

N. Ailon and Z. Rudnick, ‘Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, *Acta Arith*., **113** (2004), no. 1, 31–38 (MSN).

There have been lots of articles on this and related problems. For example, a Google search on "Ailon Rudnick gcd" brings up recent articles such as:

On some extensions of the Ailon–Rudnick theorem, A Ostafe -

*Monatshefte für Mathematik*, 2016 (MSN)On a variant of the Ailon–Rudnick theorem in finite characteristic. D Ghioca, LC Hsia, TJ Tucker -

*New York Journal of Mathematics*, 2017 (MSN)Greatest common divisors of iterates of polynomials, LC Hsia, T Tucker -

*Algebra & Number Theory*, 2017 (MSN)

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