Fast algorithm for solving superelliptic equations of certain types (Q2714421)

An algorithm is given for finding all integer solutions of the Diophantine equation NEWLINE\[NEWLINEy^2=x^{2k}+a_{2k-1}x^{2k-1}+\dots+a_1 x+a_0=F(x),NEWLINE\]NEWLINE where \(a_{2k-1},\dots,a_1,a_0\) are integers and \(k\geq 1\) is a natural number. At first a special decomposition of the polynomial \(F(x)\) is used, then one has to determine the real roots of two polynomials defining a short interval. The integer elements of this interval must be checked by computing the values of the polynomial \(F(x)\). If \(F(x)\) is a square of an integer \(y\) then a solution \((x,\pm y)\) is found. The integer solutions of another polynomial with rational coefficients give further solutions of the given Diophantine equation. In the examples there are some mistakes: \((1,\pm 5)\) is not a solution in example 2 and \((-5,\pm 17)\) is no solution in example 3.