The Michigan Mathematical Journal

On the divisibility of Fermat quotients

Jean Bourgain, Kevin Ford, Sergei Konyagin, and Igor Shparlinski
Source: Michigan Math. J. Volume 59, Issue 2 (2010), 313-328.
First Page: Show Hide
Primary Subjects: 11A37, 11L40, 11N25
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Permanent link to this document: http://projecteuclid.org/euclid.mmj/1281531459
Digital Object Identifier: doi:10.1307/mmj/1281531459
Zentralblatt MATH identifier: 05792255
Mathematical Reviews number (MathSciNet): MR2677624

References

S. Baier and L. Zhao, An improvement for the large sieve for square moduli, J. Number Theory 128 (2008), 154--174.
Mathematical Reviews (MathSciNet): MR2382775
Zentralblatt MATH: 05237771
Digital Object Identifier: doi:10.1016/j.jnt.2007.03.004
J. Bourgain and M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z^*_Q,$ where $Q$ is composite with few prime factors, Geom. Funct. Anal. 16 (2006), 327--366.
Mathematical Reviews (MathSciNet): MR2231466
Zentralblatt MATH: 1183.11047
Digital Object Identifier: doi:10.1007/s00039-006-0558-7
J. Bourgain, S. V. Konyagin, and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm, Internat. Math. Res. Notices (2008), 1--29.
Mathematical Reviews (MathSciNet): MR2439546
------, Corrigenda to: Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithm, Internat. Math. Res. Notices (2009), 3146--3147.
Zentralblatt MATH: 05604438
R. Crandall, K. Dilcher, and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), 433--449.
Mathematical Reviews (MathSciNet): MR1372002
Zentralblatt MATH: 0854.11002
Digital Object Identifier: doi:10.1090/S0025-5718-97-00791-6
R. Ernvall and T. Metsänkylä, On the $p$-divisibility of Fermat quotients, Math. Comp. 66 (1997), 1353--1365.
Mathematical Reviews (MathSciNet): MR1408373
Zentralblatt MATH: 0903.11002
Digital Object Identifier: doi:10.1090/S0025-5718-97-00843-0
W. L. Fouché, On the Kummer--Mirimanoff congruences, Quart. J. Math. Oxford Ser. (2) 37 (1986), 257--261.
Mathematical Reviews (MathSciNet): MR854625
Zentralblatt MATH: 0604.10007
Digital Object Identifier: doi:10.1093/qmath/37.3.257
A. Granville, Some conjectures related to Fermat's last theorem, Number theory (Banff, 1988), pp. 177--192, de Gruyter, New York, 1990.
Mathematical Reviews (MathSciNet): MR1106660
Zentralblatt MATH: 0702.11015
------, On pairs of coprime integers with no large prime factors, Exposition. Math. 9 (1991), 335--350.
Mathematical Reviews (MathSciNet): MR1137813
D. R. Heath-Brown and S. V. Konyagin, New bounds for Gauss sums derived from $k$th powers, and for Heilbronn's exponential sum, Quart. J. Math. Oxford Ser. (2) 51 (2000), 221--235.
Mathematical Reviews (MathSciNet): MR1765792
Zentralblatt MATH: 0983.11052
Digital Object Identifier: doi:10.1093/qjmath/51.2.221
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), 411--484.
Mathematical Reviews (MathSciNet): MR1265913
Zentralblatt MATH: 0797.11070
Y. Ihara, On the Euler--Kronecker constants of global fields and primes with small norms, Algebraic geometry and number theory, Progr. Math., 850, pp. 407--451, Birkhäuser, Boston, 2006.
Mathematical Reviews (MathSciNet): MR2263195
Zentralblatt MATH: 1185.11069
Digital Object Identifier: doi:10.1007/978-0-8176-4532-8_5
W. Keller and J. Richstein, Solutions of the congruence $a^p-1\equiv1$ $\text\rm(mod\,p^r),$ Math. Comp. 74 (2005), 927--936.
Mathematical Reviews (MathSciNet): MR2114655
Zentralblatt MATH: 1137.11301
Digital Object Identifier: doi:10.1090/S0025-5718-04-01666-7
J. Knauerand and J. Richstein, The continuing search for Wieferich primes, Math. Comp. 74 (2005), 1559--1563.
Mathematical Reviews (MathSciNet): MR2137018
Zentralblatt MATH: 1083.11006
Digital Object Identifier: doi:10.1090/S0025-5718-05-01723-0
S.V. Konyagin and C. Pomerance, On primes recognizable in deterministic polynomial time, The mathematics of Paul Erdős, vol. I, pp. 176--198, Springer-Verlag, Berlin.
Mathematical Reviews (MathSciNet): MR1425185
Zentralblatt MATH: 0869.11102
H. W. Lenstra, Miller's primality test, Inform. Process. Lett. 8 (1979), 86--88.
Mathematical Reviews (MathSciNet): MR520273
Zentralblatt MATH: 0399.10006
Digital Object Identifier: doi:10.1016/0020-0190(79)90149-2
Y. V. Malykhin, Estimates of trigonometric sums modulo $p^r,$ Mat. Zametki 80 (2006), 793--796 (Russian); English translation in Math. Notes 80 (2006), 748--752.
Mathematical Reviews (MathSciNet): MR2311594

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The Michigan Mathematical Journal