TY - JOUR
T1 - On the Diophantine Equation x^6+ky^3=z^6+kw^3
TT -
JF - IJMSI
JO - IJMSI
VL - 15
IS - 1
UR - http://ijmsi.ir/article-1-1004-en.html
Y1 - 2020
SP - 15
EP - 21
KW - Diophantine equation
KW - Elliptic curve.
N2 - 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.
M3 10.29252/ijmsi.15.1.15
ER -