[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14317/jami.2021.405

ON CONGRUENCES INVOLVING EULER POLYNOMIALS AND THE QUOTIENTS OF FERMAT

JANG, DOUK SOO (Division of Mathematics, Science, and Computers, Kyungnam University)

Publication Information

Journal of applied mathematics & informatics / v.39, no.3_4, 2021 , pp. 405-417 More about this Journal

Abstract

The aim of this paper is to provide the residues of Euler polynomials modulo p² in terms of alternating sums of like powers of numbers in arithmetical progression. Also, we establish the analogue of a classical congruence of Lehmer.

Keywords

Euler polynomials; congruences; Fermat quotient;

Citations & Related Records

Reference

1	J.W.L. Glaisher, On the residues of the sums of the inverse powers of numbers in arithmetical progression, Q. J. Math. 32 (1901), 271-305.
2	L. Carlitz, A note on Euler numbers and polynomials, Nagoya Math. J. 7 (1954), 35-43. DOI
3	L. Carlitz and J. Levine, Some problems concerning Kummer's congruences for the Euler numbers and polynomials, Trans. Amer. Math. Soc. 96 (1960), 23-37. DOI
4	G. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhangen und durch gewisse lineare Funktional-Gleichungen definiert werden, Bericht. K. Preuss. Akad. Wiss. Berlin 15 (1850), 36-42.
5	L. Euler, Institutiones Calculi Differentialis, Petersberg, 1755.
6	A. Friedmann and J. Tamarkine, Quelques formules concernent la theorie de la function [x] et des nombres de Bernoulli, J. Reine Angew. Math. 137 (1909), 146-156.
7	J.W.L. Glaisher, On the residues of r^p-1 to modulus p², p³, etc, Quart. J. 32 (1900), 1-27.
8	M.-S. Kim, On Euler numbers, polynomials and related p-adic integrals, J. Number Theory 129 (2009), 2166-2179. DOI
9	E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350-360. DOI
10	M. Lerch, Zur Theorie des Fermatschen Quotienten (a^p-1 - 1)/p = q(a), Math. Ann. 60 (1905), 471-490. DOI
11	R. Mestrovic, Congruences involving the Fermat quotient, Czechoslovak Math. J. 63 (2013), 949-968. DOI
12	J.L. Raabe, Zuruckfuhrung einiger Summen und bestmmtiem Integrale auf die Jacob-Bernoullische Function, J. Reine Angew. Math. 42 (1851) 348-367.
13	Z.-W. Sun, Introduction to Bernoulli and Euler polynomials, A Lecture Given in Taiwan on June, 2002, http://maths.nju.edu.cn/zwsun/BerE.pdf.
14	J.J. Sylvester, Sur une propriete des nombres premiers qui se ratache au theoreme de Fermat, C. R. Acad. Sci. Paris 52 (1861), 161-163.
15	S.S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III (Urbana, IL, 2000), A.K. Peters, Natick, MA, 2002.