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A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING NONPRIMITIVE PYTHAGOREAN TRIPLES

Part of: Diophantine approximation, transcendental number theory Diophantine equations

Published online by Cambridge University Press:  21 October 2020

YASUTSUGU FUJITA Open the ORCID record for YASUTSUGU FUJITA [Opens in a new window]  and
MAOHUA LE Open the ORCID record for MAOHUA LE [Opens in a new window]
YASUTSUGU FUJITA*
Affiliation:
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
MAOHUA LE
Affiliation:
Institute of Mathematics, Lingnan Normal College, Zhangjiang, Guangdong, 524048, China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Jeśmanowicz conjectured that $(x,y,z)=(2,2,2)$ is the only positive integer solution of the equation $(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$ , where n is a positive integer and f, g are positive integers such that $f>g$ , $\gcd (\kern1.5pt f,g)=1$ and $f \not \equiv g\pmod 2$ . Using Baker’s method, we prove that: (i) if $n>1$ , $f \ge 98$ and $g=1$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $x>z>y$ ; and (ii) if $n>1$ , $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)\; 2 \nmid s$ and $s<2^{r/2}$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $y>z>x$ . Thus, Jeśmanowicz’ conjecture is true if $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)$ .

Keywords

Jeśmanowicz’ conjecturePythagorean tripleternary purely exponential Diophantine equation

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 29 - 39
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Bennett, M. A., Ellenberg, J. S. and Ng, N. C., ‘The Diophantine equation ${A}^4+{2}^{\delta }{B}^2={C}^n$ ’, Int. J. Number Theory 6 (2010), 311–338.CrossRefGoogle Scholar
Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, Wiad. Mat. 1 (1955/1956), 196–202 (in Polish).Google Scholar
Laurent, M., ‘Linear forms in two logarithms and interpolation determinants II’, Acta Arith. 133 (2008), 325–348.CrossRefGoogle Scholar
Le, M.-H., ‘A note on Jeśmanowicz’ conjecture concerning Pythagorean triples’, Bull. Aust. Math. Soc. 59 (1999), 477–480.CrossRefGoogle Scholar
Le, M.-H., Scott, R. and Styer, R., ‘A survey on the ternary purely exponential Diophantine equation ${a}^x+{b}^y={c}^z$ ’, Surv. Math. Appl. 14 (2019), 109–140.Google Scholar
Lu, W.-D., ‘On the Pythagorean numbers $4{n}^2-1$ , $4n$ and $4{n}^2+1$ ’, J. Sichuan Univ. Nat. Sci. Ed. 5 (1959), 39–42 (in Chinese).Google Scholar
Miyazaki, T., ‘A remark on Jeśmanowicz’ conjecture for the non-coprimality case’, Acta Math. Sin. Engl. Ser. 31 (2015), 1255–1260.CrossRefGoogle Scholar
Mordell, L. J., Diophantine Equations (Academic Press, London, 1969).Google Scholar
Yang, H. and Fu, R.-Q., ‘A note on Jeśmanowicz’ conjecture concerning primitive Pythagorean triples’, J. Number Theory 156 (2015), 183–194.CrossRefGoogle Scholar
Zhang, X.-W. and Zhang, W.-P., ‘The exponential Diophantine equation ${(({2}^{2m}-1)n)}^x+{({2}^{m+1}n)}^y={(({2}^{2m}+1)n)}^z$ ’, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 57 (2014), 337–344.Google Scholar