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Five curious congruences modulo $p2$

In: Mathematica Slovaca, vol. 65, no. 3
Romeo Meštrović

Details:

Year, pages: 2015, 451 - 462
Keywords:
congruence modulo $p$ ($p^2$), Fermat quotient, harmonic number
About article:
Let $p≥ 3$ be a prime, and let $qp(2):=(2p-1-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note, we prove that

$$ ∑1≤ k≤ p-1, k odd((1) / (k))\binom{p-1}{k}\equiv -qp(2) (mod p2). $$

As an application, using two combinatorial identities due to T. B. Staver in 1947 and G. Galperin and H. Gauchman in 2004, we obtain four curious combinatorial congruences modulo $p2$. As an auxiliary result, here we present an elementary proof of a congruence established by E. Lehmer in 1938. Notice that this congruence together with some other congruences given in our lemmas leads to the elementary proof of a beautiful Morley's congruence due to F. Morley in 1895.
How to cite:
ISO 690:
Meštrović, R. 2015. Five curious congruences modulo $p2$. In Mathematica Slovaca, vol. 65, no.3, pp. 451-462. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0033

APA:
Meštrović, R. (2015). Five curious congruences modulo $p2$. Mathematica Slovaca, 65(3), 451-462. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0033
About edition:
Published: 1. 6. 2015