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Sum of powers of near-factorials

Source: Kvant Magazine No. 8 2020 M2616

March 9, 2023
number theoryDivisibilityKvant

Problem Statement

Let p>5p>5 be a prime number. Prove that the sum ((p−1)!1)p+((p−1)!2)p+⋯+((p−1)!p−1)p\left(\frac{(p-1)!}{1}\right)^p+\left(\frac{(p-1)!}{2}\right)^p+\cdots+\left(\frac{(p-1)!}{p-1}\right)^pis divisible by p3p^3.
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