Computing all S-integral points on elliptic curves

Abstract

Let the elliptic curve E be defined by the equationformula herewith a1, …, a6 ∈ ℤ. Define a finite set of places S = {q1, …, qs−1, qs = ∞} of ℚ and put Q = max {q1, …, qs−1}. Let E(ℚ) denote the set of (x, y) ∈ ℚ2 satisfying (1) and the infinite point [Oscr ].The multiplicative height of a rational point P = (x, y) ∈ E(ℚ) is defined as the following product over all places q of ℚ (including q = ∞):formula herewhere the [mid ]x[mid ]qs are the normalized multiplicative absolute values of ℚ corresponding to the places q.

ON THE DIOPHANTINE EQUATION x2 + C = 2yn

“…For the cases n = 3 and n = 4, we show that (8) can be reduced to computing S-integral points on a handful of elliptic curves. The problem can now be solved by applying standard algorithms for computing S-integral points on elliptic curves (see, for example, [26]). Fortunately these algorithms are available as an inbuild functions in the computer package MAGMA [10].…”

A modular approach to the generalized Ramanujan-Nagell equation

“…A practical method for the explicit computation of all S-integral points on a Weierstrass elliptic curve has been developed by Pethő, Zimmer, Gebel and Herrmann in [21] and has been implemented in Magma [5]. The relevant routine SIntegralPoints worked without problems for all triples (i, j, k) except for (i, j, k) ∈ {(5, 2, 96), (5, 1, 120), (5, 2, 156), (5, 2, 180), (5,2,192), (5, 2, 220), (3, 1, 232), (5, 0, 232), (5, 2, 232), (5, 0, 240), (5, 2, 240), (5, 2, 244), (5, 0, 304), (5, 1, 304), (5, 2, 304), (3, 2, 316), (5, 0, 316), (5, 2, 316), (5, 2, 324), (5, 0, 360), (5, 1, 364), (5, 2, 364), (3, 2, 372), (5, 1, 372), (5, 2, 372), (5, 2, 376), (3, 1, 412), (3, 2, 412), (5, 0, 412), (5, 0, 420), (5, 0, 432), (5, 1, 432), (3, 2, 444), (5, 1, 444), (5, 2, 444), (5, 0, 456), (5, 1, 456), (5, 2, 460), (5, 1, 492), (5, 1, 516), (5, 2, 516), (3, 1, 520), (5, 0, 520), (5, 2, 520), (5, 2, 532), (5, 1, 544), (5, 2, 552), (3, 2, 612), (5, 0, 612), (5, 1, 612), (5, 2, 612), (5, 1, 616), (5, 0, 640), (5, 2, 640), (3, 2, 652), (5, 2, 652), (5, 2, 660), (3, 2, 664)(5, 0, 664), (5, 2, 664), (5, 1, 684), (5, 0, 700), (5, 1, 700), (5, 2, 700), (5, 0, 712), (5, 0, 720), (5, 1, 720)}.…”

On the Diophantine equation $x^2 + 7^\alpha \cdot 11^\beta = y^n$

“…where S D f7; 11g with the numerator of Y being coprime to 77; in view of the restriction gcd.x; y/ D 1: Now we need to determine all the f7; 11g-integral points on the above 36 elliptic curves. At this stage we note that in [33] Pethő, Zimmer, Gebel and Herrmann developed a practical method for computing all S Integral points on Weierstrass elliptic curve and their method has been implemented in MAGMA [11] as a routine under the name SIntegralPoints.The subroutine SIntegralPoints of MAGMA worked without problems for all .˛1;ˇ1/ except for .˛1;ˇ1/ D .5; 5/. MAGMA determined the appropriate Mordell-Weil groups except this case and we deal with this exceptional case separately.…”