## Data— joint with Rafael von Känel — | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

## Contents- Elliptic curves with good reduction outside S
- S-unit equation
- Mordell equation
- Cubic Thue equation
- Cubic Thue–Mahler equation
- Generalized Ramanujan–Nagell equation
- Sums of units being a square or a cube
- S-integral points on elliptic curves of given conductor
- S-integral points on elliptic curves of high rank
This page contains the data attached to the paper "Solving S-unit, Mordell, Thue, Thue–Mahler and generalized Ramanujan–Nagell equations via Shimura–Taniyama conjecture". We put all data under a Creative commons 3.0 by-nc license. In what follows primes and integers are always rational.
## Elliptic curves with good reduction outside SLet S be a finite set of primes. We are interested in describing explicitly the set M(S) of all elliptic curves over ℚ up to ℚ-isomorphism with good reduction outside S. Each elliptic curve E in M(S) is represented via the two invariants c _{4}, c_{6} ∈ ℤ of some (and thus any) minimal Weierstrass model of E over the integers.
In particular, the elliptic curve E has a Weierstrass equation
y
^{2} = x^{3} − 27c_{4}x − 54c_{6}.
Table 1: The sets M(S) for certain finite sets S of primes.
Here N
Parts of our data were already computed by Cremona–Lingham, Koutsianas and Bennett–Rechnitzer,
see for example [vKM, Section 4.2.5] for a discussion of known methods computing M(S).
Further, the special case N Our data motivates the following question. Let n be a positive integer and let S(n) denote the set of the first n primes. Question 1: For all sets S of primes with |S| ≤ n, it holds |M(S)| ≤ |M(S(n))|? The answer is affirmative for all sets S of primes (and their subsets) appearing in the above table. In light of this we conjecture in [vKM, Section 4.2.5] that Question 1 has a positive answer in general, up to adding some absolute constant.
## Data for S-unit equationsLet S be a finite set of primes. Rational numbers with numerator and denominator having all prime factors in S are called S-units.
We consider the S-unit equation
x + y = 1,
where x and y are S-units.
If c denotes the least common denominator of x and y, then we can write x = a/c and y = b/c with a, b, c coprime integers which are S-units and satisfy a + b = c.
Thus solving the S-unit equation is equivalent to solving
a + b = c,
where a, b, c are coprime integers which are S-units.
To break the natural symmetry of the solutions to this equation,
we may and do assume that 0 < a ≤ b < c.
For certain sets S, the following Tables 1, 2, 3 list the set of solutions and their number, up to symmetry.
Table 1: Solutions when S is the set of the first n primes, for 1 ≤ n ≤ 16:
Here the cases n = 1, ..., 6 were already computed by de Weger; see for example [vKM, Section 3] for a discussion of known methods solving S-unit equations. The number N(S) of solutions of the S-unit equation is either zero or 6N'(S) - 3, where N'(S) is the number of solutions up to symmetry. Note that in Tables 1, 2, 3 we display the numbers N'(S). Let n be a positive integer and recall that S(n) denotes the set of the first n primes. Our data motivates the following Question 2: For all sets S of primes with |S| ≤ n, it holds N(S) ≤ N(S(n))? The answer is affirmative for each set S and their subsets appearing in our database. In light of this we conjecture in [vKM, Section 3.2.6] that Question 2 has a positive answer up to adding some absolute constant.
Table 2: Solutions for all sets S of primes with ∏p∈Sp ≤ 10
^{k}, for 1 ≤ k ≤ 7:
An abc-triple is given by positive coprime integers a, b, c with a + b = c, and the radical of an integer m is the square-free product of the primes dividing m.
The above tables contain in particular all abc-triples with radical at most 10
Table 3: Solutions when S consists of the first 5 Fermat primes together with 2, or the first 6 Mersenne primes together with 2:
OpenOffice Calc: In case you would like to use the data with OpenOffice Calc, simply open the .txt with this program, and at the start-up dialog add ":+=()" to "Separator options->Separated by->Others", and set "From row" to 11.
## Data for Mordell equationsLet S be a finite set of primes. An S-integer is a rational number whose denominator has all its prime factors in S.
Let a be a nonzero integer and consider the Mordell equation
y
where x and y are S-integers.
The Mordell equation defines an affine model of an elliptic curve over ℚ whose ℚ-rational points form a finitely generated abelian group.
In what follows we shall refer by ^{2} = x^{3} + a,
rank to the rank of this group.
In the following Table 1, S denotes the set of the first n primes.
Table 1: S-integral points on Mordell curves with bounded |a|:
All Mordell curves with |a| ≤ 10000 have rank at most 4, and thus the table contains all their S-integral points for n = 300. Similarly, all Mordell curves with |a| ≤ 10 have rank 0 or 1, and thus the table contains all their S-integral points for n = 100000. Fueter gave an explicit formula for the rational points on Mordell curves of rank 0. In particular the above list for rank 0 is well known and stated only for the sake of completeness. In the important special case when S is empty, our data was already computed by Gebel–Pethő–Zimmer and Bennett–Ghadermarzi. See for example [vKM, Section 4] for a discussion of known methods solving Mordell equations. Our algorithm depends on having a Mordell–Weil basis of the ℚ-rational points of the elliptic curve associated to the Mordell equation. The following sage-object file contains a list of all such bases for |a| ≤ 10000. It is based on a database computed by Gebel–Pethő–Zimmer, with a few updates (a basis for a = 7823 was missing, and the given points for a = -7086 and -6789 were not saturated; according to some computations in magma and sage this list is correct now, unconditionally). In the following Table 2, S denotes the set of the first n primes.
Table 2: S-integral points on some Mordell curves of high rank:
For a given Mordell equation, our data suggests that the number of S-integral solutions grows very slowly in terms of the cardinality |S|. In view of this we propose the following conjecture.
Conjecture: There is a constant c, depending only on a, such that for all finite sets S of primes the Mordell equation y In [vKM, Section 4.2.7] we motivate and discuss various questions and conjectures which moreover refine the above conjecture.
## AppendixSome old data, which is now superseded by the above data. We say that an integral solution (x, y) of Mordell's equation is primitive if the greatest common divisor of x^{3} and y^{2} is sixth power-free.
Suppose that our given integer a has the prime factorization a = ±∏p^{α(p)} and then define r_{2}(a) = ∏p^{min(2,α(p))}.
The file
contains all primitive solutions for each a with r
## Data for cubic Thue equationsLet S be a finite set of primes, and let a, b, c, d be integers. Suppose that m is a nonzero integer and consider the cubic Thue equation
ax
where x and y are S-integers. Here we assume that the discriminant D of f(x, y) is nonzero, where f(x, y) is the binary form given by the left hand side of the equation; see for example [Salmon, p. 175] for the definition of D.
^{3} + bx^{2}y + cxy^{2} + dy^{3} = m,
Table 1: S-integral solutions of all cubic Thue equations f(x, y) = 1 with nonzero discriminant D bounded by |D| ≤ D
_{max}, where S is the set of the first n primes.
The above table only contains the data for reduced f(x, y), which is sufficient to cover all cubic binary forms with nonzero discriminant D bounded by |D| ≤ D
Table 2: S-integral solutions of certain classical cubic Thue equations f(x, y) = m with 1 ≤ m ≤ m
_{max}, where S is the set of the first n primes.
Here we consider only positive m in order to break the symmetry f(−x, −y) = −f(x, y). See for example [vKM, Section 5] for a discussion of known methods solving cubic Thue equations. For a given cubic Thue equation, our data suggests that the number of S-integral solutions grows very slowly in terms of the cardinality |S|. In view of this we propose the following conjecture.
Conjecture: There is a constant c, depending only on D and m, such that for all finite sets S of primes the cubic Thue equation f(x, y) = m of discriminant D has at most (|S| + 2) In [vKM, Sections 5.3.2 and 4.2.7] we motivate and discuss various questions and conjectures which moreover refine the above conjecture.
## Data for cubic Thue–Mahler equationsLet S be a finite set of primes, and let a, b c, d be integers. We consider the cubic Thue–Mahler equation
ax
where x, y, z are integers and z has all prime factors in S.
Here we assume again that the discriminant D of f(x, y) is nonzero, where f(x, y) is the binary form given by the left hand side of the equation.
We say that a solution (x, y, z) of the cubic Thue–Mahler equation is primitive if the integers x and y are coprime.
^{3} + bx^{2}y + cxy^{2} + dy^{3} = z,
Table 1: Primitive solutions of all cubic Thue–Mahler equations f(x, y) = z with nonzero discriminant D bounded by |D| ≤ D
_{max}, where S is the set of the first n primes.
The above table only contains the data for reduced f(x, y), which is sufficient to cover all cubic binary forms with nonzero discriminant D bounded by |D| ≤ D
Table 2: Primitive solutions of certain classical cubic Thue–Mahler equations.
Here the solutions of the first three equations were already known: The first equation is due to Agrawal–Coates–Hunt–van der Poorten, the second one to Tzanakis–de Weger, and the third one to a further paper of Tzanakis–de Weger. See for example [vKM, Section 5] for a discussion of known methods solving cubic Thue–Mahler equations. In the case (a, b, c, d) = (0, 1, 1, 0), the equation becomes xy(x+y) = z and thus the cubic Thue–Mahler equation is in this case equivalent to the S-unit equation.
## Data for generalized Ramanujan–Nagell equationsLet S be a finite set of primes, and let b be a nonzero integer. We consider the following generalized Ramanujan–Nagell equation
x
where x is an S-integer and y is an S-unit.
This generalizes the original Ramanujan–Nagell equation x^{2} + b = y,
^{2} + 7 = 2^{k} solved in positive integers x and k.
Table 1: Solutions of all generalized Ramanujan–Nagell equations with nonzero b bounded by |b| ≤ b
_{max}, where S is the set of the first n primes.
We next consider two classical cases of the generalized Ramanujan–Nagell equation which are both inspired by the original problem.
Table 2: Solutions of the Diophantine problem x
^{2} + 7 = y, where x, y are integers and y has all prime factors among the first n primes.
Table 3: Solutions of the Diophantine problems x
^{2} + 7 = d^{k} for any integer d in the range 2 ≤ d ≤ d_{max}, where x and k are positive integers.
We point out that certain cases in the above tables are already known by the work of Pethő–de Weger. See for example [vKM, Section 6] for a discussion of known methods solving generalized Ramanujan–Nagell equations.
## Data for sums of units being a square or a cubeLet S be a finite set of primes and let ℓ be in {2, 3}. We consider the equation
x + y = z
where x, y, z are integers and x, y are S-units (i.e. all primes dividing xy lie in S).
For certain sets S, we now list all solutions up to symmetry.
^{ℓ},
Table 1 (Squares): For certain sets S, all solutions of x + y = z
^{2} up to symmetry.
Table 2 (Cubes): For certain sets S, all solutions of x + y = z
^{3} up to symmetry.
Parts of the above data were already computed by de Weger and Bennett–Billerey. See for example [vKM, Section 6.2.1] for a discussion of known methods computing sums of units which are squares or cubes.
## S-integral points on elliptic curves of given conductorLet S be a finite set of primes, and let E be an elliptic curve over ℚ. We are interested to find all S-integral points on E, that is we would like to determine all S-integral solutions of a minimal Weierstrass equation of E over the integers. Cremona's database contains in particular all elliptic curves over ℚ of conductor at most 1000. For certain finite sets S of primes, we used our elliptic logarithm sieve to compute all S-integral points on these elliptic curves.
Table 1: All S-integral points on all elliptic curves over ℚ with conductor N ≤ N
_{max}, where S is the set of the first n primes.
The input of our elliptic logarithm sieve requires an initial height bound and a Mordell–Weil basis of the ℚ-rational points of E: We applied the initial height bound of Pethő–Zimmer–Gebel–Herrmann based on the theory of logarithmic forms and we used the Mordell–Weil bases contained in Cremona's database. We point out that the data in the above table is already known in the important special case when S is the empty set; indeed Cremona computed all integral points. See for example [vKM, Sections 4 and 11] for a discussion of known methods enumerating all S-integral points on elliptic curves over ℚ. For a given elliptic curve over ℚ, our data suggests that the number of S-integral points grows very slowly in terms of the cardinality |S|. In view of this we propose the following conjecture.
Conjecture:
There is a constant c, depending only E, such that for all finite sets S of primes the elliptic curve E has at most (|S| + 2) In [vKM, Sections 11.11.3 and 4.2.7] we motivate and discuss various questions and conjectures which moreover refine and generalize the above conjecture.
## S-integral points on elliptic curves of high rankLet E be an elliptic curve over ℚ. The ℚ-rational points E(ℚ) of E form a finitely generated abelian group of rank r. We now consider some curves with large rank r. To determine all S-integral points on these curves, we applied our elliptic logarithm sieve with the initial height bounds of Pethő–Zimmer–Gebel–Herrmann based on the theory of logarithmic forms and the Mordell-Weil bases described below.
E
y
^{2} + xy = x^{3} − 5818216808130x + 5401285759982786436.
To compute all S-integral points on E
Table 1: All S-integral points on E
_{Kr}, where S is the set of the first n primes.
E
y
^{2} + y = x^{3} − 6349808647 x + 193146346911036.
Mestre also found 12 independent ℚ-rational points on E
Assumption: The rank of E
Under this assumption, Siksek determined a Mordell–Weil basis of E
Table 2: Under the above assumption, this table lists all S-integral points on E
_{Me}, where S is the set of the first n primes.
Next we consider curves E for which the rank r of E(ℚ) is even higher. Unfortunately, we could not determine a Mordell–Weil bases for E(ℚ) when r was larger than 12. To illustrate the efficiency of our elliptic logarithm sieve, we computed at least the S-integral points on E which lie in the subgroup of E(ℚ) generated by the ℚ-rational torsion points of E and r independent ℚ-rational points of E; see Table 3.
E
y
^{2} = x^{3} + x^{2} − 1692310759026568999140789578145x + 839379398840982294584587970773038145228669599.
E
y
^{2} + xy + y = x^{3} − 957089489055751752507625259831765957846101x + 351598252970651757672333752869879740192822872602430248013582348.
E
y
^{2} + xy + y = x^{3} − x^{2} + 31368015812338065133318565292206590792820353345x + 302038802698566087335643188429543498624522041683874493555186062568159847.
Table 3: All S-integral points on E which lie in the subgroup of E(ℚ) generated by the ℚ-rational torsion points of E and r independent ℚ-rational points of E,
where S is the set of the first n primes and E is one of the above three curves.
E
y
^{2} + xy + y = x^{3} − x^{2} − 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429.
Table 4: All integral points on E
_{El28} which lie in the subgroup of E_{El28}(ℚ) generated by 28 independent ℚ-rational points. E_{El28}(ℚ) has trivial torsion.
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