Let be a prime number and . Recall that we conjectured in this post that
is divisible by . From exercise 2 of this post we know that factors as into prime ideals in the ring . So . Therefore the right-hand side of is divisible by . So we are off by at most one factor of ! I believe this approach can be improved upon to account for the extra factor, since we haven’t used any property of the sum.
for any . So our best result thus far is the following:
Theorem (weaker version of Fleck’s).
Goal: Improve the floors to ceilings.