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

**Fact. **.

*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.