Glasnik Matematicki, Vol. 54, No. 2 (2019), 271-277.

AN OPEN PROBLEM ON JEŚMANOWICZ' CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES

Hai Yang and Ruiqin Fu

School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, P.R. China
e-mail: xpuyhai@163.com

School of Science, Xi'an Shiyou University, Xi'an, Shaanxi, 710065, P.R. China
e-mail: xsyfrq@163.com

Abstract.   Let m>31 be an even integer with gcd(m,31)=1. In this paper, using some elementary methods, we prove that the equation (m2-312)x+(62m)y=(m2+312)z has only the positive integer solution (x,y,z)=(2,2,2). This result resolves an open problem raised by T. Miyazaki (Acta Arith. 186 (2018), 1-36) about Jeśmanowicz' conjecture concerning primitive Pythagorean triples.

2010 Mathematics Subject Classification.   11D61

Key words and phrases.   Ternary purely exponential Diophantine equation, Jeśmanowicz' conjecture, primitive Pythagorean triple, elementary method

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.2.02

References:

  1. R. D. Carmichael, On the numerical factors of arithmetic forms α n ± β n, Ann. of Math. (2) 15 (1913/1914), 49-70.
    MathSciNet     CrossRef

  2. L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat. (2) 1 (1955/1956), 196-202.
    MathSciNet    

  3. W. D. Lu, On Pythagorean numbers 4n2-1, 4n and 4n2+1, J. Sichuan Univ. Nat. Sci. 5 (1959), 39-42.

  4. T. Miyazaki, Contributions to some conjectures on a ternary exponential Diophantine equation, Acta Arith. 186 (2018), 1-36.
    MathSciNet     CrossRef

  5. L. J. Mordell, Diophantine equations, Academic Press, London-New York, 1969.
    MathSciNet    

  6. G. Soydan, M. Demirci, I. N. Cangül and A. Togbé, On the conjecture of Jeśmanowicz, Int. J. Appl. Math. Stat. 56 (2017), 46-72.
    MathSciNet    

  7. N. Terai, On Jeśmanowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory. 141 (2014), 316-323.
    MathSciNet     CrossRef
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