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1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} = p + (p-1)! \mod p^2

Source: 11th QEDMO p6 Seniors (6-8. 12. 2012) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 16, 2021
number theoryremainderSum of powers

Problem Statement

Let pp be an odd prime number. Prove that 1p−1+2p−1+...+(p−1)p−1≡p+(p−1)!mod  p21^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2
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