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(-1)^{ (p+1)/2 }+\sum {2n \choose n}c^n is divisible by p

Source: 7th QEDMO problem 9 (14. - 17. 1. 2010) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 9, 2021
number theorydivisible

Problem Statement

Let pp be an odd prime number and cc an integer for which 2c−12c -1 is divisible by pp. Prove that (−1)p+12+∑n=0p−12(2nn)cn(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n is divisible by pp.
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