A robust implementation for solving the $S$-unit equation and several applications

Abstract

Let K be a number field, and S a finite set of places in K containing all infinite places. We present an implementation for solving the S-unit equation x + y = 1, x, y ∈ O × K,S in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets S. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant where 2 is totally ramified. In addition, we improve bounds for S-unit equation solutions that would allow enumeration of a certain class of genus 2 curves, and use the implementation to find all solutions to some cubic Ramanujan-Nagell equations.1 See also the recent translation [16] by Fuchs.

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“…Exceptional units and exceptional S-units (which allow both x and 1−x to be S-units) remain of substantial practical interest because of a wide variety of applications to number theory and other fields. These include: enumerating elliptic/Fermat curves over K with good reduction outside a fixed set of primes [14,16,23]; understanding finitely generated groups, arithmetic graphs, and recurrence sequences [7]; and many Diophantine problems [12], including asymptotic versions of Fermat's last theorem [1,9]. See [8] for many more applications.…”

“…For instance, [18] and [20] study the number of exceptional units in fields of degree 3 and 4. Over low-degree number fields, there has also been recent progress on computing the set of solutions to general S-unit equations, building on a long tradition of methods involving Baker's method and/or the theory of linear forms in real, complex, and p-adic logarithms [1].…”

Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees

“…3 and 5 mod 2 3 , which are both the forms a number can take if they are also 1 mod 2 2 . Further, we use blue squares to showcase when we know the exact valuation for a given classification of x, and red circles for when the valuation is still not known exactly, but we do know that it's greater than or equal to the value in the circle.…”

“…We fix the following notations: The cubic equation (1.2) has a similar rich history of being studied. For y = 1, Beukers [4] has proven that (1.2) has at most five solutions in x ∈ Z. Alvarado et al [2,Theorem 1.1] analyzed the cubic Ramanujan-Nagell equation x 3 + D = q n for prime q > 3 and n, k > 0, and where D = 3 k . Letting q be a prime and 3 < q ≤ 500, they prove and list all integer solutions to the equation, and they claim that their method can also be used to find the integer solutions to the equation x 3 + p k = q n where p, q are different odd primes.…”

On the unit equation over cyclic number fields of prime degree

“…Moreover, de Weger [17] has given a rather efficient algorithm for determining the solutions to (1.1) which combines Baker's bounds for linear forms in logarithms with the LLL algorithm. De Weger's algorithm has since been refined by a number of authors, for example [1], [8], [15].…”