Congruences involving the Fermat quotient

Romeo Meštrović

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 949-968
  • ISSN: 0011-4642

Abstract

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Let p > 3 be a prime, and let q p ( 2 ) = ( 2 p - 1 - 1 ) / p be the Fermat quotient of p to base 2 . In this note we prove that ∑ k = 1 p - 1 1 k · 2 k ≡ q p ( 2 ) - p q p ( 2 ) 2 2 + p 2 q p ( 2 ) 3 3 - 7 48 p 2 B p - 3 ( mod p 3 ) , which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that q p ( 2 ) 3 ≡ - 3 ∑ k = 1 p - 1 2 k k 3 + 7 16 ∑ k = 1 ( p - 1 ) / 2 1 k 3 ( mod p ) , which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ∑ k = 1 p - 1 1 / ( k 2 · 2 k ) modulo p 2 that also generalizes a related Sun’s congruence modulo p .

How to cite

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Meštrović, Romeo. "Congruences involving the Fermat quotient." Czechoslovak Mathematical Journal 63.4 (2013): 949-968. <http://eudml.org/doc/260793>.

@article{Meštrović2013,
abstract = {Let $p>3$ be a prime, and let $q_p(2)=(2^\{p-1\}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that \[ \sum \_\{k=1\}^\{p-1\} \frac\{1\}\{k\cdot 2^k\} \equiv q\_p(2)-\frac\{pq\_p(2)^2\}\{2\}+ \frac\{p^2 q\_p(2)^3\}\{3\} -\frac\{7\}\{48\} p^2 B\_\{p-3\}\hspace\{10.0pt\}(\@mod \; p^3), \] which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that \[ q\_p(2)^3 \equiv -3\sum \_\{k=1\}^\{p-1\} \frac\{2^k\}\{k^3\}+ \frac\{7\}\{16\} \sum \_\{k=1\}^\{(p-1)/2\} \frac\{1\}\{k^3\} \hspace\{10.0pt\}(\@mod \; p), \] which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum _\{k=1\}^\{p-1\}1/(k^2\cdot 2^k)$ modulo $p^2$ that also generalizes a related Sun’s congruence modulo $p$.},
author = {Meštrović, Romeo},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fermat quotient; $n$th harmonic number of order $m$; Bernoulli number; Fermat quotient; Bernoulli number},
language = {eng},
number = {4},
pages = {949-968},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Congruences involving the Fermat quotient},
url = {http://eudml.org/doc/260793},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Meštrović, Romeo
TI - Congruences involving the Fermat quotient
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 949
EP - 968
AB - Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that \[ \sum _{k=1}^{p-1} \frac{1}{k\cdot 2^k} \equiv q_p(2)-\frac{pq_p(2)^2}{2}+ \frac{p^2 q_p(2)^3}{3} -\frac{7}{48} p^2 B_{p-3}\hspace{10.0pt}(\@mod \; p^3), \] which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that \[ q_p(2)^3 \equiv -3\sum _{k=1}^{p-1} \frac{2^k}{k^3}+ \frac{7}{16} \sum _{k=1}^{(p-1)/2} \frac{1}{k^3} \hspace{10.0pt}(\@mod \; p), \] which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum _{k=1}^{p-1}1/(k^2\cdot 2^k)$ modulo $p^2$ that also generalizes a related Sun’s congruence modulo $p$.
LA - eng
KW - Fermat quotient; $n$th harmonic number of order $m$; Bernoulli number; Fermat quotient; Bernoulli number
UR - http://eudml.org/doc/260793
ER -

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