A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS FOR PURE POWERS

Abstract

In this paper, we prove that there does not exist a set of 11 polynomials with coefficients in a field of characteristic 0, not all constant, with the property that the product of any two distinct elements plus 1 is a perfect square. Moreover, we prove that there does not exist a set of 5 polynomials with the property that the product of any two distinct elements plus 1 is a perfect kth power with k ≥ 7. Combining these results, we get an absolute upper bound for the size of a set with the property that the product of any two elements plus 1 is a pure power.