Mordell’s equation: a classical approach

Abstract

We solve the Diophantine equation Y 2 = X 3 + k for all nonzero integers k with |k| 10 7 . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

Sums of two cubes as twisted perfect powers, revisited

“…and Ẽ is one of the following : curve a 2 a 4 ∆ 36q.2.a1 −3d 3q −2 4 3 3 q 2 36q.2.a2 6d −3 2 8 3 3 q 36q.2.b1 9d 3 3 q −2 4 3 9 q 2 36q.2.b2 −18d −3 3 2 8 3 9 q…”

“…for odd positive integers b 2 , d 6 and b 6 , so that (7.1) d 2 6 = 4 · 3 b2 − 3 b6 + 16. In general, this equation has precisely the solutions 1, 1, 3), (5, 1, 1), (11,3,1) and (31,5,3) in odd positive integers; none of these correspond to a prime values of q > 73. To prove this, note that an elementary argument easily yields that b 2 > b 6 unless |d 6 | ≤ 5.…”

“…(of Lemma 5.1) Writing x = 3x 1 + x 0 for x 0 ∈ {0, 1, 2}, we have that a solution to the equation 2 x = 3y 2 + 5 necessarily corresponds to an integer point on the (Mordell) elliptic curve Y 2 = X 3 − 2 2x0 · 3 3 · 5 (with Y = 2 x0 · 3 2 · y and X = 3 · 2 x0+x1 ). We can find the integer points for each of these curves at http://www.math.ubc.ca/~bennett/BeGa-data.html (see [3] for more details), whereby the stated conclusion obtains.…”

On cubic polynomials with a given discriminant

“…533-539] going back to the work of Bachet of 1621. From the extensive literature concerning (1.1) see, for example, [1,[6][7][8] and, [17]. There is a standard method for computing all integer solutions of (1.1) using David's bounds and lattice reduction.…”

A parametrised family of Mordell curves with a rational point of order 3

“…has been subjected to extensive investigations ( [1], [2], [3], [4], [6], [7], [8], [10], [11,Chapter 26,, [12], [16]). Eq.…”