A NOTE ON THE DIOPHANTINE EQUATION
2013 ◽
Vol 89
(2)
◽
pp. 316-321
◽
AbstractLet $(a, b, c)$ 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.
2016 ◽
Vol 95
(1)
◽
pp. 5-13
◽
2020 ◽
Vol 4
(2)
◽
pp. 103
2010 ◽
Vol 81
(2)
◽
pp. 177-185
◽
Keyword(s):
2012 ◽
Vol 08
(03)
◽
pp. 813-821
◽
2014 ◽
Vol 90
(1)
◽
pp. 9-19
◽
2013 ◽
Vol 94
(1)
◽
pp. 50-105
◽
2009 ◽
Vol 05
(06)
◽
pp. 1117-1128
◽
Keyword(s):
2006 ◽
Vol 02
(02)
◽
pp. 195-206
◽
Keyword(s):