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A NOTE ON THE DIOPHANTINE EQUATION $\mathop{(na)}\nolimits ^{x} + \mathop{(nb)}\nolimits ^{y} = \mathop{(nc)}\nolimits ^{z} $

Part of: Diophantine equations

Published online by Cambridge University Press:  07 August 2013

MOU JIE DENG
MOU JIE DENG*
Affiliation:
Department of Applied Mathematics, Hainan University, Haikon, Hainan 570228, PR China email Moujie_Deng@163.com
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Abstract

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Let $(a, b, c)$ be a primitive Pythagorean triple satisfying ${a}^{2} + {b}^{2} = {c}^{2} . $ In 1956, Jeśmanowicz conjectured that for any given positive integer $n$ the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $x= y= z= 2. $ In this paper, for the primitive Pythagorean triple $(a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)$ with $k= {2}^{s} $ for some positive integer $s\geq 0$, we prove the conjecture when $n\gt 1$ and certain divisibility conditions are satisfied.

Keywords

Diophantine equationsPythagorean triplesJeśmanowicz’ conjecture

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 316 - 321
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Dem´janenko, V. A., ‘On Jeśmanowicz’ problem for Pythagorean numbers’, Izv. Vyss. Ucebn. Zaved. Mat. 48 (1965), 52–56 (in Russian).Google Scholar
Deng, M. J., ‘On the Diophantine equation $\mathop{(15n)}\nolimits ^{x} + \mathop{(112n)}\nolimits ^{y} = \mathop{(113n)}\nolimits ^{z} $’, J. Natur. Sci. Heilongjiang University 24 (2007), 617–620 (in Chinese).Google Scholar
Deng, M. and Cohen, G. L., ‘On the conjecture of Jeśmanowicz concerning Pythagorean triples’, Bull. Aust. Math. Soc. 57 (1998), 515–524.Google Scholar
Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, Wiad. Mat. 1 (1955–1956), 196–202 (in Polish).Google Scholar
Le, M. H., ‘A note on Jeśmanowicz’ conjecture concerning Pythagorean triples’, Bull. Aust. Math. Soc. 59 (1999), 477–480.Google Scholar
Lu, W. T., ‘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).Google Scholar
Miyazaki, T., ‘On the conjecture of Jeśmanowicz concerning Pythagorean triples’, Bull. Aust. Math. Soc. 80 (2009), 413–422.Google Scholar
Miyazaki, T., ‘Generalizations of classical results on Jeśmanowicz’ conjecture concerning Pythagorean triples’, J. Number Theory 133 (2013), 583–595.CrossRefGoogle Scholar
Sierpiński, W., ‘On the equation ${3}^{x} + {4}^{y} = {5}^{z} $’, Wiad. Mat. 1 (1955–1956), 194–195 (in Polish).Google Scholar
Tang, M. and Yang, Z. J., ‘Jeśmanowicz’ conjecture revisited’, Bull. Aust. Math. Soc., to appear; available online, doi:10.1017/S0004972713000038.Google Scholar
Yang, Z. J. and Tang, M., ‘On the Diophantine equation $\mathop{(8n)}\nolimits ^{x} + \mathop{(15n)}\nolimits ^{y} = \mathop{(17n)}\nolimits ^{z} $’, Bull. Aust. Math. Soc. 86 (2012), 348–352.CrossRefGoogle Scholar