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## Colloquium Mathematicum

2000 | 86 | 1 | 25-30
Tytuł artykułu

### A note on a conjecture of Jeśmanowicz

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let a, b, c be relatively prime positive integers such that $a^2+b^2=c^2$. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of $(an)^x+(bn)^y=(cn)^z$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation $(u^2-v^2)^x+(2uv)^y=(u^2+v^2)^z$, for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
25-30
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-14
Twórcy
autor
• Heilongjiang Nongken Teachers' College, A Cheng City, People's Republic of China
autor
• School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia
Bibliografia
• [1] J. R. Chen, On Jeśmanowicz' conjecture, Acta Sci. Natur. Univ. Szechan 2 (1962), 19-25 (in Chinese).
• [2] V. A. Dem'janenko [V. A. Dem'yanenko], On Jeśmanowicz' problem for Pythagorean numbers, Izv. Vyssh. Uchebn. Zaved. Mat. 48 (1965), 52-56 (in Russian).
• [3] M. Deng and G. L. Cohen, On the conjecture of Jeśmanowicz concerning Pythagorean triples, Bull. Austral. Math. Soc. 57 (1998), 515-524.
• [4] L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat. 1 (1955/56), 196-202 (in Polish).
• [5] C. Ko, On the Diophantine equation $(a^2-b^2)^x+(2ab)^y=(a^2+b^2)^z$, Acta Sci. Natur. Univ. Szechan 3 (1959), 25-34 (in Chinese).
• [6] M. H. Le, A note on Jeśmanowicz' conjecture concerning Pythagorean numbers, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 97-98.
• [7] W. T. Lu, On the Pythagorean numbers $4n^2-1$, 4n and $4n^2+1$, Acta Sci. Natur. Univ. Szechuan 2 (1959), 39-42 (in Chinese).
• [8] W. Sierpiński, On the equation $3^x+4^y=5^z$, Wiadom. Mat. 1 (1955/56), 194-195 (in Polish).
• [9] K. Takakuwa, On a conjecture on Pythagorean numbers. III, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 345-349.
• [10] K. Takakuwa, A remark on Jeśmanowicz' conjecture, ibid. 72 (1996), 109-110.
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