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This paper presents an algorithm that, given an integer n > 1, finds the largest integer k such that n is a kth power. A previous algorithm by the first author took time b 1+o(1) where b = lg n; more precisely, time b exp(O(√ lg b lg lg b)); conjecturally, time b(lg b) O(1). The new algorithm takes time b(lg b) O(1). It relies on relatively complicated… (More)

In a recent paper I established an analogue of the Lindemann-Weierstrass part of Ax-Schanuel for the elliptic modular function. Here I extend this to include its first and second derivatives. A generalisation is given that includes exponential and Weierstrass elliptic functions as well.

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