While working on a project, I have run into a situation where I have integers x and n so that $x^n \equiv (x+1)^n \equiv (x+2)^n$ mod $p$ for a prime $p$. It seems to me that this an extremely restrictive condition, and I was wondering if there are any results about when (or if?) it can happen, but I couldn't figure out what to search. Any thoughts? What if I add the additional restriction that they are also congruent to $(x+3)^n$, etc.


Thanks!



Successive nth powers mod p?
Write $a = (x+1)/x, b = (x+2)/x$, then your condition is equivalent to $a^n \equiv b^n \equiv 1 \mod p$ and $2a-b \equiv 1 \mod p$. Now, without loss of generality, you can assume $n|(p-1)$ (otherwise replace $n$ by the gcd of $n$ and $p-1$). Write $m = (p-1)/n$ and $a=u^m,b=v^m$. Finally, your conditions become $2u^m-v^m=1$. This defines a curve and, by the Weil bound, it will have points with $uv \ne 0$ if $p \gg m^4$. So, assuming $n|(p-1)$ what you need is $n \gg p^{3/4}$ to guarantee solutions. On the other hand, if $n$ is small, typically there won't be any solutions. If you add the $x+3$ condition, you still get a curve, but of higher genus, so a similar thing will happen with different bounds.


