Differences between perfect powers : the Lebesgue-Nagell Equation

Abstract

We develop a variety of new techniques to treat Diophantine equations of the shape x 2 + D = y n , 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 y n > x 2 and x 2 − y n has no prime divisor exceeding 11.

$\mathbb{Q}$-curves and the Lebesgue-Nagell equation

“…It is worth observing that equation ( 4) is the more problematical case (in comparison to equation (3)), for the purposes of application of bounds for linear forms in logarithms. In case of equation ( 3), results of Bugeaud [6] imply that n < 4.5 • 10 6 q 2 log 2 q, which we can, with care, sharpen to an upper bound upon n of somewhat less than 10 6 for, say, q = 3 in equation (3). Even with such a bound, it remains impractical to finish the problem via this approach, since we have no reasonable techniques to obtain a contradiction for a fixed value of n in (3), while, as discussed in [4, pp.…”

“…solution with y = ±1, valid for all (odd) exponents n. The analogous obstruction, in case of equation (3) with q ∈ {41, 73, 89}, or equation ( 4), for q ∈ {17, 41, 89, 97}, is slightly more subtle, arising from the fact that q ± 8 is square, in the first case, and from the identities (5) 23 2 − 17 = 2 9 , 13 2 − 41 = 2 7 , 91 2 − 89 = 2 13 and 15 2 − 97 = 2 7 , in the second.…”

“…In recent papers of the first-and third-named authors [3] and [4], various tools are developed to tackle equation (1) in the two "difficult" cases, where either D > 0 and y is even, or where D < 0. In particular, [4] focusses upon these situations where, additionally, it is assumed that D has a single prime divisor.…”