Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-04T19:07:35.111Z Has data issue: false hasContentIssue false
  • English
  • Français

On the conjecture of Jesmanowicz concerning Pythagorean triples

Published online by Cambridge University Press:  17 April 2009

Moujie Deng  and
G.L. Cohen
Moujie Deng
Affiliation:
Heilongjiang Nongken Teachers’ College, A Cheng CityPeople's Republic of China
G.L. Cohen
Affiliation:
School of Mathematical SciencesUniversity of Technology, SydneyPO Box 123Broadway NSW 2007Australia, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let a, b, c be relatively prime positive integers such that a2 + b2 = c2. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an)x + (bn)y = (en)z in positive integers is x = y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > 1, c = b + 1 and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples (a, b, c) = (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) and (11, 60, 61).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 3 , June 1998 , pp. 515 - 524
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Dem'janenko, V.A., ‘On Jeśmanowicz' problem for Pythagorean numbers’, (in Russian), Izv. Vysšh. Učebn. Zaved. Mat. 48 (1965), 52–56.Google Scholar
[2]Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, (in Polish), Wiadom. Mat. 1 (1955/1956), 196–202.Google Scholar
[3]Maohua, L., ‘A note on Jeśmanowicz' conjecture’, Colloq. Math. 69 (1995), 47–51.Google Scholar
[4]Maohua, L., ‘On Jeśmanowicz' conjecture concerning Pythagorean numbers’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 97–98.Google Scholar
[5]Lu, W.T., ‘On the Pythagorean numbers 4n 2 − 1, 4n and 4n 2 + 1’, (in Chinese), Acta sci. Natur. Univ. Szechuan 2 (1959), 39–42.Google Scholar
[6]Sierpiński, W., ‘On the equation 3x + 4y = 5z’, (in Polish), Wiadom. Mat. 1 (1955/1956), 194–195.Google Scholar
[7]Takakuwa, K., ‘On a conjecture on Pythagorean numbers. III’, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 345–349.Google Scholar
[8]Takakuwa, K., ‘A remark on Jeśmanowicz' conjecture’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 109–110.CrossRefGoogle Scholar
[9]Terai, N., ‘The Diophantine equation a x + b y = c z. II’, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 109–110.CrossRefGoogle Scholar