Integer points on 𝑦²=𝑥³-7𝑥+10

Abstract

It is apparent that the groups of rational points on curves (0.1)-(0.4) will probably have several generators. Using the arithmetic of the field Q(p), p3 -7p + 10 = 0 (see Section 1), one can perform the standard 2-descents (see Cassels [6] or Birch and Swinnerton-Dyer [4]) on the curve (0.1) to determine that its rational Mordell-Weil group actually has rank 2, but the details are omitted. It seems plausible that the rank of the curves (0.2), (0.3), (0.4) is in each instance equal to 4, but this has not been specifically verified.It is perhaps also worth noting here the integer point on (0.4) given by (x, y) = (1645085185,66724078854865) because of the large size of the coordinates; Lang [11], [12] has made some interesting conjectures on the size of integral points on the curve y2 = x3 + ax + b, in particular that for an integral point (x0, y0) then |x0|« max(|a|3,|èp)* for some uniform k. The curve (0.1) has conductor 23 • 83 (see Täte [19] for a recipe on calculation of the conductor), rank 2, and 26 integer points, none of which, however, is particularly Targe'. Examples possessing a sizeable solution include:(i) the curve j2 = x3 -x + 1 of conductor 22 • 23, rank 1, and 12 integer points including (x, y) = (56,419). (This is the curve denoted 92c in the tables of Birch and Kuyk [3]; see also the review [MR 82g: 10037] by the first author of a paper by Sansone[16].)