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Common divisors of a^n-1 and b^n-1 over function fields

Joseph H. Silverman
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Abstract

Ailon and Rudnick have shown that if $a,b \in C[T]$ are multiplicatively independent polynomials, then $°(\gcd(a^n-1,b^n-1))$ is bounded for all $n\ge1$. We show that if instead $a,b \in F[T]$ for a finite field $F$ of characteristic $p$, then $°(\gcd(a^n-1,b^n-1))$ is larger than $Cn$ for a constant $C=C(a,b)>0$ and for infinitely many $n$, even if $n$ is restricted in various reasonable ways (e.g., $gec(n,p)=1$).

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Joseph H. Silverman. 2004-01-26. Common divisors of a^n-1 and b^n-1 over function fields. https://arxiv.org/abs/math/0401356

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