# Greatest common divisor of $(a^n+1,b^n+1)$

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

• "Sufficient to prove that" is an interesting formulation :) If it is taken literally, then I am clueless about the answer since nobody has been able to prove this. In other words, this is a well known conjecture. However, using the work of Corvaja and Zannier based on Schmidt's Subspace theorem, one can at least prove the sub-exponential upper bound $\exp(\varepsilon n)$ on this gcd, for all $\varepsilon > 0$, and infinitely many $n$. (Indeed all large enough $n$ if additionally $a$ and $b$ are multiplicatively independent.) Jul 31, 2019 at 8:27
• If you can prove there is one n, you should be able to prove there are infinitely many. Gerhard "Start With A Simpler Case" Paseman, 2019.07.31. Jul 31, 2019 at 8:33
• Is it at least true that there are infinitely many values of $n$ with $(2^n+1,3^n+1)=1$?
– Seva
Jul 31, 2019 at 8:51
• Thank you very much for enlightening me. I was almost sure that it is a famous conjecture, but I couldn't find any papers about it. Jul 31, 2019 at 9:17
• Is it known that for any set $P$ of primes of positive relative density ($|P\cap[1,x]\gg x/\log x$) there are infinitely many exponents $n$ such that $b^n+1$ is divisible only by the primes from $P$? Choosing $P$ to be the set of all primes $p\equiv 3\pmod 4$ such that $a$ is a quadratic residue modulo $p$, for every such $n$ we will have $(a^n+1,b^n+1)=1$ since $a^n+1$ is not divisible by primes $p\in P$ in view of $a^n\equiv -1\pmod p$.
– Seva
Jul 31, 2019 at 12:41

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).