# A note on a conjecture of Jeśmanowicz

Colloquium Mathematicae (2000)

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

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topDeng, 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

top- [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. Zbl0916.11020
- [4] L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat. 1 (1955/56), 196-202 (in Polish).
- [5] C. Ko, On the Diophantine equation ${({a}^{2}-{b}^{2})}^{x}+{\left(2ab\right)}^{y}={({a}^{2}+{b}^{2})}^{z}$, 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. Zbl0876.11013
- [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] W. Sierpiński, On the equation ${3}^{x}+{4}^{y}={5}^{z}$, Wiadom. Mat. 1 (1955/56), 194-195 (in Polish). Zbl0074.27204
- [9] K. Takakuwa, On a conjecture on Pythagorean numbers. III, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 345-349. Zbl0822.11025
- [10] K. Takakuwa, A remark on Jeśmanowicz' conjecture, ibid. 72 (1996), 109-110. Zbl0863.11025

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