Abstract
This paper begins with a short historical survey on Catalan's equation, namely xp-yq=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 32-23=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification.
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References
J.W.S. Cassels, “On the equation a x -b y = 1, II,” Proc. Camb. Phil. Soc. 56, (1960) 97-103.
K. Inkeri, “On Catalan's problem,” Acta Arith. 9, (1964) 285-290.
K. Inkeri, “On Catalan's conjecture,” J. Number Th. 34, (1990) 142-152.
Ko Chao, “On the diophantine equation x 2 = y n + 1, xy≠ 0,” Sci. Sinica, 14, (1965) 457-460.
M. Langevin, “Quelques applications de nouveaux résultats de van der Poorten,” Sém. Delange-Pisot-Poitou, 1977/78, Paris, Exp. 4.
M. Laurent, M. Mignotte, Y. Nesterenko, “Formes linéaires en deux logarithmes et déterminants d'interpolation,” J. Numb. Th., 55, (1995) 285-321.
V.A. Lebesgue, “Sur l'impossibilité en nombres entiers de l'équation x m = y 2 + 1,” Nouv. Ann. Math., 9, (1850) 178-181.
M. Mignotte, “Un critère élémentaire pour l'équation de Catalan,” C.R. Math. Rep. Acad. Sci. Canada, 15, no 5, (1993) 199-200.
M. Mignotte, “A criterion on Catalan's equation,” J. Numb. Th., 52, (1995) 280-284.
M. Mignotte and Y. Roy, “Catalan's equation has no new solution with either exponent less than 10651” Experimental Mathematics, 4, (1995) 259-268.
T. Nagell, “Des équations indéterminées x 2 +x+1 = y n et x 2 +x+1 = 3yn,” Nordsk. Mat. Forenings Skr. (1), 2, (1920).
P. Ribenboim, Catalan's conjecture, Acad. Press, Boston, 1994.
W. Schwarz, “A note on Catalan's equation,” Acta Arith., 72, (1995) 277-279.
R. Tijdeman, “On the equation of Catalan,” Acta Arith., 29, (1976) 197-209.
L.C. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York, 1982.
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Mignotte, M., Roy, Y. Lower Bounds for Catalan's Equation. The Ramanujan Journal 1, 351–356 (1997). https://doi.org/10.1023/A:1009701725510
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DOI: https://doi.org/10.1023/A:1009701725510