A new result on Jeśmanowicz' conjecture

Gökhan Soydan (Uludag University of Bursa)
Pondělí 7. října 2019, 14:20 hodin
Zasedací místnost DFP, 4. patro budovy G, kampus Husova (Univerzitní nám. 1410/1)

KOMIX 7-10-19

[Pozvánka v PDF]

Abstract

Let m, n be positive integers such that m > n, gcd(m,n) = 1 and m ≠ n (mod 2). In 1956, L. Jeśmanowicz' [1] conjectured that the equation has only the positive integer solution (x,y,z) = (2,2,2). This conjecture has been still unsolved. For over twenty years, many papers have investigated Jeśmanowicz' conjecture for the case that (mod 4). In this talk, ombining a lower bound for linear forms in two logarithms due to M. Laurent [2] with some elementary methods in number theory, we prove that if (mod 4) and > 30.8 , then Jeśmanowicz' conjecture is true [5]. This result improves some previous results [3], [4] and [6].

[1] L. Jeśmanowicz', Several remarks on Pythagorean numbers, Wiadom. Math. 1 (1955/1956), 196–202 (in Polish).
[2] M. Laurent, Linear forms in two logarithms and interpolation determinants II, Acta Arith., 133 (2008), 325–348.
[3] M. H. Le, On Jeśmanowicz' conjecture concerning Pythagorean numbers, Proc. Japan Acad., Ser. A, 72, (1996), 97–98.
[4] T. Miyazaki and N. Terai, On Jeśmanowicz' conjecture concerning Pythagorean triples II, Acta Math. Hung. 147, (2015), 286–293.
[5] M. H. Le and G. Soydan, An application of Baker's method to the Jeśmanowicz' conjecture on primitive Pythagorean triples, Periodica Math. Hung. (2019), to appear.
[6] P. Z. Yuan and Q. Han, Jeśmanowicz' conjecture and related equations, Acta Arith. 184, (2018), 37–49.

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