Catalan’s equation 𝑥^{𝑝}-𝑦^{𝑞}=1 and related congruences

Aaltonen, M.; Inkeri, K.

doi:10.1090/S0025-5718-1991-1052082-7

Catalan’s equation $x^ p-y^ q=1$ and related congruences
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by M. Aaltonen and K. Inkeri PDF

Math. Comp. 56 (1991), 359-370 Request permission

Abstract:

We investigate solutions of Catalan’s equation ${x^p} - {y^q} = 1$ in nonzero integers x, y, p, q. By use of class numbers and congruences ${p^q} \equiv p\;\pmod {q^2}$ we show the impossibility of the equation for a large number of prime exponents p, q. Both theoretical and computer results are included. We also study lower bounds on possible, hitherto unknown, solutions $x,y > 2$; we especially wish to communicate the bound $x,y \geq {10^{500}}$.

References

A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 213–222. MR 0314736

On the Diophantine equation

14

Theorematum quorundam arithmeticorum demonstrationes

M. Gut, Abschätzungen für die Klassenzahlen der quadratischen Körper, Acta Arith. 8 (1962/63), 113–122 (German). MR 184926, DOI 10.4064/aa-8-2-113-122
Seppo Hyyrö, Über das Catalansche Problem, Ann. Univ. Turku. Ser. A I 79 (1964), 10 (German). MR 179127
K. Inkeri, On Catalan’s problem, Acta Arith. 9 (1964), 285–290. MR 168518, DOI 10.4064/aa-9-3-285-290
K. Inkeri, On Catalan’s conjecture, J. Number Theory 34 (1990), no. 2, 142–152. MR 1042488, DOI 10.1016/0022-314X(90)90145-H
Wells Johnson, On the nonvanishing of Fermat quotients $(\textrm {mod}$ $p)$, J. Reine Angew. Math. 292 (1977), 196–200. MR 450193, DOI 10.1515/crll.1977.292.196
Wells Johnson, On the $p$-divisibility of the Fermat quotients, Math. Comp. 32 (1978), no. 141, 297–301. MR 463091, DOI 10.1090/S0025-5718-1978-0463091-4
Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456

Quelques applications de nouveaux résultats de van der Porten

^e

Sur l’impossibilité en nombres entiers de l’equation

9

Des équations indéterminées

et

2

Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767

Groupe des classes d’idéaux des corps quadratiques imaginaires

Kettenbrüche

P. Ribenboim, Consecutive powers, Exposition. Math. 2 (1984), no. 3, 193–221. MR 783135
Paulo Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080, DOI 10.1007/978-1-4684-9938-4
Hans Riesel, Note on the congruence $a^{p-1}=1$ $(\textrm {mod}$ $p^{2})$, Math. Comp. 18 (1964), 149–150. MR 157928, DOI 10.1090/S0025-5718-1964-0157928-6
T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR 891406, DOI 10.1017/CBO9780511566042
R. Tijdeman, On the equation of Catalan, Acta Arith. 29 (1976), no. 2, 197–209. MR 404137, DOI 10.4064/aa-29-2-197-209
B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325–367. MR 901244, DOI 10.1016/0022-314X(87)90088-6