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.

Furthermore,

for any . So our best result thus far is the following:

**Theorem (weaker version of Fleck’s). **

.

Goal: Improve the floors to ceilings.

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