Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$? 


The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not exist, see [here](http://kvant.mccme.ru/1984/08/p34.htm "Kvant 1984").
But discussion here 


[On the exact reference of a cute Diophantine problem](https://mathoverflow.net/questions/255785/on-the-exact-reference-of-a-cute-diophantine-problem/255792?noredirect=1#comment630273_255792)


suggests that it should be very hard to prove. Maybe, a counterexample may be bound? Roughly speaking, a probability of this event is about $1/p^2$, for each $p$ we have about $p$ events (even for $b=1$, I was previously wrong that it may be supposed without loss of generality, thanks for Noam Elkies for noting this) and so as $\sum 1/p=\infty$, we may expect them (and even infinitely many!) But this series converges very slowly, so the minimal example may be large.



May $p^3$ divide $(a+b)^p-a^p-b^p$?
There are apparently lots of examples. The smallest is $a=1$, $b=2$, and $p=7$.


