2021/09/19 by Bennett, Michael A., Siksek, Samir · 1 citation
#11D61 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2109.09128
We develop a variety of new techniques to treat Diophantine equations of the shape x2+D =yn, based upon bounds for linear forms in p-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers x and y, and n ≥ 3, with the property that yn > x2 and x2-yn has no prime divisor exceeding 11.