# A note on a conjecture of Jeśmanowicz

Colloquium Mathematicae (2000)

• Volume: 86, Issue: 1, page 25-30
• ISSN: 0010-1354

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## Abstract

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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 ${\left(an\right)}^{x}+{\left(bn\right)}^{y}={\left(cn\right)}^{z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${\left({u}^{2}-{v}^{2}\right)}^{x}+{\left(2uv\right)}^{y}={\left({u}^{2}+{v}^{2}\right)}^{z}$, 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.

## How to cite

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Deng, Moujie, and Cohen, G.. "A note on a conjecture of Jeśmanowicz." Colloquium Mathematicae 86.1 (2000): 25-30. <http://eudml.org/doc/210838>.

@article{Deng2000,
abstract = {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.},
author = {Deng, Moujie, Cohen, G.},
journal = {Colloquium Mathematicae},
keywords = {exponential Diophantine equation; Jeśmanowicz’s conjecture},
language = {eng},
number = {1},
pages = {25-30},
title = {A note on a conjecture of Jeśmanowicz},
url = {http://eudml.org/doc/210838},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Deng, Moujie
AU - Cohen, G.
TI - A note on a conjecture of Jeśmanowicz
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 25
EP - 30
AB - 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.
LA - eng
KW - exponential Diophantine equation; Jeśmanowicz’s conjecture
UR - http://eudml.org/doc/210838
ER -

## References

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1. [1] J. R. Chen, On Jeśmanowicz' conjecture, Acta Sci. Natur. Univ. Szechan 2 (1962), 19-25 (in Chinese).
2. [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. [3] M. Deng and G. L. Cohen, On the conjecture of Jeśmanowicz concerning Pythagorean triples, Bull. Austral. Math. Soc. 57 (1998), 515-524. Zbl0916.11020
4. [4] L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat. 1 (1955/56), 196-202 (in Polish).
5. [5] C. Ko, On the Diophantine equation ${\left({a}^{2}-{b}^{2}\right)}^{x}+{\left(2ab\right)}^{y}={\left({a}^{2}+{b}^{2}\right)}^{z}$, Acta Sci. Natur. Univ. Szechan 3 (1959), 25-34 (in Chinese).
6. [6] M. H. Le, A note on Jeśmanowicz' conjecture concerning Pythagorean numbers, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 97-98. Zbl0876.11013
7. [7] W. T. Lu, On the Pythagorean numbers $4{n}^{2}-1$, 4n and $4{n}^{2}+1$, Acta Sci. Natur. Univ. Szechuan 2 (1959), 39-42 (in Chinese).
8. [8] W. Sierpiński, On the equation ${3}^{x}+{4}^{y}={5}^{z}$, Wiadom. Mat. 1 (1955/56), 194-195 (in Polish). Zbl0074.27204
9. [9] K. Takakuwa, On a conjecture on Pythagorean numbers. III, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 345-349. Zbl0822.11025
10. [10] K. Takakuwa, A remark on Jeśmanowicz' conjecture, ibid. 72 (1996), 109-110. Zbl0863.11025

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