On the p-divisibility of Fermat quotients

The authors carried out a numerical search for Fermat quotients Q a = (a p-1 - 1)/p vanishing mod p, for 1 ≤ a ≤ p - 1, up top < 10 6 . This article reports on the results and surveys the associated theoretical properties of Q a . The approach of fixing the prime p rather than the base a leads to some aspects of the theory apparently not published before.

[1]  D. R. Heath-Brown.  An estimate for Heilbronn's exponential sum , 1996 .

[2]  Charles Helou,et al.  Norm residue symbol and cyclotomic units , 1995 .

[3]  Tauno Metsänkylä,et al.  Cyclotomic invariants for primes between 125000 and 150000 , 1991 .

[4]  P. Ribenboim.  The new book of prime number records , 1996 .

[5]  Wells Johnson,et al.  On the /^-Divisibility of the Fermât Quotients , 2010 .

[6]  Maurice Mignotte,et al.  Catalan's Equation Has No New Solution with Either Exponent Less Than 10651 , 1995, Exp. Math..

[7]  Wolfgang Schwarz.  A note on Catalan's equation , 1995 .

[8]  M. Aaltonen,et al.  Catalan's equation xp−yq=1 and related congruences , 1991 .

[9]  L. Dickson.  History of the Theory of Numbers , 1924, Nature.

[10]  H. Riesel.  Note on the Congruence a p-1 ≡1 (mod p 2 ) , 1964 .

[11]  Jiro Suzuki.  On the generalized Wieferich criteria , 1994 .

[12]  H. S. Vandiver.  An aspect of the linear congruence with applications to the theory of Fermat’s quotient , 1915 .

[13]  H. Riesel.  Note on the congruence ^{-1}=1 ( ²) , 1964 .

[14]  Takashi Agoh.  On the Kummer-Mirimanoff congruences , 1990 .

[15]  Peter L. Montgomery.  New solutions of ^{-1}≡1\pmod{²} , 1993 .

[16]  Richard E. Crandall,et al.  Irregular primes and cyclotomic invariants to four million , 1993 .

[17]  P. L. Montgomery.  New solutions of ap-1 ≡ 1 (mod p2) , 1993 .

[18]  Andrew Granville,et al.  Refining the conditions on the Fermat quotient , 1985, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  M. Aaltonen,et al.  Catalan’s equation ^{}-^{}=1 and related congruences , 1991 .

[20]  Karl Dilcher,et al.  A search for Wieferich and Wilson primes , 1997, Math. Comput..

[21]  Hendrik W. Lenstra,et al.  Miller's Primality Test , 1979, Inf. Process. Lett..

[22]  W. B..  History of the Theory of Numbers , Nature.

[23]  Richard Mollin,et al.  Some Conjectures Related to Fermat's Last Theorem , 1990 .

[24]  Emma Lehmer,et al.  On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson , 1938 .

[25]  Wells Johnson.  On the nonvanishing of Fermat quotients (mod p). , 1977 .

[26]  R. Tijdeman.  On the equation of Catalan , 1976 .

[27]  M. Lerch.  Zur Theorie des Fermatschen Quotienten $$\frac{{a^{p - 1} - 1}}{p} = q(a).$$ , 1905 .

[28]  Don Coppersmith.  Fermat’s last theorem (case 1) and the Wieferich criterion , 1990 .