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Difference of polynomials is not squarefree for odd primes

Source: Polish Second Round 2001

March 6, 2012
algebrapolynomialmodular arithmeticnumber theory proposednumber theory

Problem Statement

Let k,n>1k,n>1 be integers such that the number p=2k−1p=2k-1 is prime. Prove that, if the number (n2)−(k2)\binom{n}{2}-\binom{k}{2} is divisible by pp, then it is divisible by p2p^2.
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