In the last post we showed that
and then we used Lucas’ theorem to evaluate the sum modulo .
My initial attempt at evaluating the sum was to note that
where , but then I was stuck. Numerical examples suggested that this sum is in fact divisible by , but I could not think of a way of extending the argument from the last post. Nonetheless, some extensive google search recently revealed that
Theorem (Fleck, 1913). For any ,
So taking gives the desired claim after noting that
Proof. Write with . If then the equation is just , and if then it is .
This article contains a proof of Fleck’s result using the identity . Unfortunately I don’t have a good grasp of the theory behind it.