Finding integers k for which a given Diophantine equation has no solution in kth powers of integers

@article{Granville1992FindingIK,
title={Finding integers k for which a given Diophantine equation has no solution in kth powers of integers},
author={A. Granville},
journal={Acta Arithmetica},
year={1992},
volume={60},
pages={203-212}
}

(1) f(x1 , x k 2 , . . . , x k n) = 0 has solutions in non-zero integers x1, x2, . . . , xn. For homogeneous diagonal f of degree one, Davenport and Lewis [DL] showed that k ∈ T (f) whenever (n − 1) ≥ k ≥ 18; however, Ankeny and Erdős [AE] showed that T (f) has zero density in the set of all positive integers provided that all distinct subsets of the set of coefficients of f have different sums. For general polynomials f , Ribenboim [Ri] showed that certain values of k cannot belong to T (f… CONTINUE READING