Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE DIOPHANTINE EQUATION (an)x + (bn)y = (cn)z

  • MA, MI-MI (SCHOOL OF MATHEMATICAL SCIENCES AND INSTITUTE OF MATHEMATICS NANJING NORMAL UNIVERSITY) ;
  • WU, JIAN-DONG (SCHOOL OF MATHEMATICAL SCIENCES AND INSTITUTE OF MATHEMATICS NANJING NORMAL UNIVERSITY)
  • Received : 2014.05.16
  • Published : 2015.07.31
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Abstract

In 1956, $Je{\acute{s}}manowicz$ conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,\;b,\;c)=(4k^2-1,\;4k,\;4k^2+1)$.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. Demjanenko, On Jesmanowicz' problem for Pythagorean numbers, Izv. Vyss. Ucebn. Zaved. Mat. 48 (1965), no. 5, 52-56.
  2. M. J. Deng, A note on the Diophantine equation $(na)^x$ + $(nb)^y$ = $(nc)^z$, Bull. Austral. Math. Soc. 89 (2014), no. 2, 316-321. https://doi.org/10.1017/S000497271300066X
  3. M. J. Deng and G. L. Cohen, On the conjecture of Jesmanowicz concerning Pythagorean triples, Bull. Austral. Math. Soc. 57 (1998), no. 3, 515-524. https://doi.org/10.1017/S0004972700031920
  4. L. Jesmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat. 1 (1955/56), 196-202.
  5. M. H. Le, A note on Jesmanowicz conjecture, Colloq. Math. 69 (1995), no. 1, 47-51.
  6. M. H. Le, On Jesmanowicz conjecture concerning Pythagorean numbers, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 5, 97-98. https://doi.org/10.3792/pjaa.72.97
  7. M. H. Le, A note on Jesmanowicz' conjecture concerning Pythangorean triples, Bull. Austral. Math. Soc. 59 (1999), no. 3, 477-480. https://doi.org/10.1017/S0004972700033177
  8. W. T. Lu, On the Pythagorean numbers $4n^2$-1, 4n and $4n^2$+1, Acta Sci. Natur. Univ. Szechuan 2 (1959), 39-42.
  9. T. Miyazaki, Generalizatioins of classical results on Jesmanowicz' conjecture concerning Pythagorean triples, J. Number Theory 133 (2013), no. 2, 583-595. https://doi.org/10.1016/j.jnt.2012.08.018
  10. W. Sierpinski, On the Diophantine equation $3^x$ + $4^y$ = $5^z$, Wiadom. Mat. 1 (1955/56), 194-195.
  11. K. Takakuwa, On a conjecture on Pythagorean numbers. III, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 9, 345-349. https://doi.org/10.3792/pjaa.69.345
  12. K. Takakuwa, A remark on Jesmanowicz' conjecture, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 6, 109-110. https://doi.org/10.3792/pjaa.72.109
  13. K. Takakuwa and Y. Asaeda, On a conjecture on Pythagorean numbers. II, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 8, 287-290. https://doi.org/10.3792/pjaa.69.287
  14. M. Tang and J. X. Weng, Jesmanowicz' conjecture with Fermat numbers, Taiwanese J. Math. 18 (2014), no. 3, 925-930. https://doi.org/10.11650/tjm.18.2014.3942
  15. M. Tang and Z. J. Yang, Jesmanowicz' conjecture revisited, Bull. Austral. Math. Soc. 88 (2013), no. 3, 486-491. https://doi.org/10.1017/S0004972713000038
  16. Z. J. Yang and M. Tang, On the Diophantine equation $(8n)^x$ + $(15n)^y$ = $(17n)^z$, Bull. Austral. Math. Soc. 86 (2012), no. 2, 348-352. https://doi.org/10.1017/S000497271100342X