%0 Journal Article
%A Shabani-Solt., H.
%A Yusefnejad, N.
%A Janfada, A. S.
%T On the Diophantine Equation x^6+ky^3=z^6+kw^3
%J Iranian Journal of Mathematical Sciences and Informatics
%V 15
%N 1
%U http://ijmsi.ir/article-1-1004-en.html
%R 10.29252/ijmsi.15.1.15
%D 2020
%K Diophantine equation, Elliptic curve.,
%X Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is not greater than 500. We exhibit a collection of (probably infinitely many) rational numbers k for which this Diophantine equation is satisfied. Finally, appealing these observations, we conjecture that the above result is true for all rational numbers k.
%> http://ijmsi.ir/article-1-1004-en.pdf
%P 15-21
%& 15
%!
%9 Research paper
%L A-10-2361-1
%+ Urmia University
%G eng
%@ 1735-4463
%[ 2020