The identity 7^3n+7^(3n+1)=(2∙7^n )^3 and its generalizations

Abstrakt

Starting with the identity 7^(3)}+7^(3n+1)=(2∙ 7^n)^3 and its sibling, we prove that for any positive integer $m$, the diophantine equation x^n+x^(n+k)=z^m has infinitely many solutions in nonzero integers x, z, n and k. We show that in case k>1 the solutions come from the Catalan's Conjecture. We also solve three similar diophantine equations.