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Some curious congruences modulo primes

preprint2010arXivOpen access
Li-Lu ZhaoZhi-Wei Sun
math.COmath.NT
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Abstract

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence: ∑0<i1<...<in<p(i1/3)(−1)i1/(i1...in)=0(modp).\sum_{0<i_1<...<i_n<p}(i_1/3)(-1)^{i_1}/(i_1...i_n)=0 (mod p).

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