AU - Shabani-Solt., H.
AU - Yusefnejad, N.
AU - Janfada, A. S.
TI - On the Diophantine Equation x^6+ky^3=z^6+kw^3
PT - JOURNAL ARTICLE
TA - IJMSI
JN - IJMSI
VO - 15
VI - 1
IP - 1
4099 - http://ijmsi.ir/article-1-1004-en.html
4100 - http://ijmsi.ir/article-1-1004-en.pdf
SO - IJMSI 1
ABĀ - 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.
CP - IRAN
IN - Department of Mathematics, Urmia University, Urmia 57561-51818, IRAN
LG - eng
PB - IJMSI
PG - 15
PT - Research paper
YR - 2020