On the Diophantine equation \(x^4-q^4 = py^3\)
Let \(p\) and \(q\) be primes. The authors consider the Diophantine equation \[ x^4-q^4=py^3 \] and prove that for coprime integers \(x\) and \(y\) no solution exists. This is done by showing that a solution to this Diophantine equation corresponds to an integer point on certain elliptic curves. But no integer point of these elliptic curves yields a solution to the original Diophantine equation. Moreover, the authors also discuss the case of \(x\) and \(y\) have a common divisor. In this case the original Diophantine equation reduces to the equation \[ x^4-1=pq^2y^3. \] Under some technical restrictions on \(p\) and \(q\) the authors show that this Diophantine equation has no solution with \(y\not =0\) and conjecture that there exist only finitely many positive integer solutions \((x,y,p,q)\) with \(p,q\) prime. In this case the authors reduce the Diophantine equation to certain Thue equations.
scientific article; zbMATH DE number 5760104