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 $(a^n+1,b^n+1)=1$ ?



Greatest common divisor of $(a^n+1,b^n+1)$
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 $b^k-1$](https://doi.org/10.4064/aa113-1-3), *Acta Arith*., **113** (2004), no. 1, 31–38 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2046966)).


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](https://mathscinet.ams.org/mathscinet-getitem?mr=3539944))
* [On a variant of the Ailon–Rudnick theorem in finite characteristic](http://nyjm.albany.edu:8000/j/2017/23_213.html). D Ghioca, LC Hsia, TJ Tucker - *New York Journal of Mathematics*, 2017 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=3625452))
* [Greatest common divisors of iterates of polynomials](https://doi.org/10.2140/ant.2017.11.1437), LC Hsia, T Tucker - *Algebra & Number Theory*, 2017 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=3687102))


