Author: Dr. Jonathan Kenigson, FRSA
Fermat’s Last Theorem is perhaps the most prominent example of a Diophantine equation, which is a polynomial equation with integer (whole number) coefficients that is to be solved over the integers and not a larger domain [1]. For instance, one may inquire which integers satisfy the equation x+2y=13 – a task that often conjures prime numbers, factorization, and abstract algebra [2]. The field is named in honor of the Greek-African mathematician Diophantus the Alexandrian, who lived in the 3rd century AD and wrote a series of books called Arithmetica [3]. During the Islamic Golden Age, mathematicians such as al-Khwarizmi, al-Karaji, and al-Haytham developed methods for solving linear and quadratic Diophantine equations and progressed towards solving higher degree equations [4]. In the 18th and 19th centuries, Euler [5], Lagrange [6], Legendre [7], and Gauss [8] studied Diophantine equations using modular arithmetic and the rudiments of the modern theory of algebraic factorization over finite fields. In the 20th century, Hilbert [9], Taniyama [10], and Shimura [10] advanced the field using algebraic geometry, modular forms, elliptic curves, and Galois theory. Roughly, Galois theory is the study of where certain polynomial functions “vanish” (or become zero) [11]. These points form algebraic structures in their own right, called Galois groups [12]. Also speaking crudely, elliptic curves are graphs of certain cubic equations (e.g. equations of degree 3) [13]. Andrew Wiles’s proof of Fermat’s Last Theorem relies heavily upon Taniyama and Shimura’s work. The Taniyama-Shimura conjecture states that every elliptic curve over the rational numbers (fractions as we typically use them) is a modular form [10]. Wiles’ proof of Taniyama-Shimura implies Fermat’s Last Theorem [14].
The first step in Wiles’ proof is to establish Taniyama-Shimura for a special class of elliptic curves that are called “semistable” [15]. Semistability is a condition that defines the curve’s behavior at prime numbers. When studying elliptic curves over number fields, it’s common to extend the base field to include all the algebraic numbers that are needed to define the curve [16]. In this setting, a prime p is said to be a prime of bad reduction if the reduction of the curve modulo p is not a smooth curve [17][18]. For example, if the coefficients have a common factor, then the equation defines a curve that is not smooth and therefore not an elliptic curve in the usual sense. A semistable elliptic curve is one that has good reduction at all but finitely many primes, and at the bad primes, the reduction is ‘as simple as possible’, meaning that the singularities are not mathematically intractable [17]. In particular, a semistable elliptic curve has only nodes or cusps as singularities at the bad primes [17]. Wiles’ proof technique exploits Galois representations and modular forms [10]. Fermat’s Last Theorem follows for all exponents except 4, a case that was later resolved by Wiles and Taylor [19]. By using modular forms in cryptography, Diophantine equations are used to create secure encryption schemes. In coding theory, they are used to design error-correcting codes [20]. Diophantine equations also reciprocally enrich group theory via the representations of groups of rational points on algebraic curves [21].
As an example, The Mordell-Weil Theorem describes the structure of the group of rational points on an elliptic curve [22]. Roughly, the Theorem states that the group of rational points on an elliptic curve over a number field (a finite extension of the rational numbers) is a finitely generated abelian group [22]. In other words, it says that the set of rational points on an elliptic curve is generated by a finite number of points. It allows one to determine whether a given Diophantine equation has a solution in rational numbers by studying the rational points on the associated elliptic curve [22]. The idea is to first find a set of generators for the group of rational points on the elliptic curve, and then use these generators to construct all rational solutions to the Diophantine equation [22]. Beside Mordell-Weil, another foundational result, Siegel’s Theorem, gives an upper bound on the number of integer solutions of a Diophantine equation in terms of the degree of the equation [23]. This trajectory of thought began in the early 20th century as Carl Ludwig Siegel concerned himself with bounding the number of integral (that is, whole-number) solutions to certain types of Diophantine equations that Gauss had discussed in terms of pure number theory [24]. Modern formulations of Seigel exploit the topological properties of the curve genus, a metric for the complexity of a curve. For an algebraic curve C of genus g > 0 defined over the field of rational numbers, there are only finitely many integral points on C. In other words, if the curve has a genus greater than 0, there are only a finite number of points on the curve with integer coordinates that lie on the graph [25]. The proof of Siegel’s Theorem relies on techniques from algebraic geometry, geometry of numbers, and the theory of Diophantine approximation [26]. The original proof by Siegel was quite involved, but later refinements and simplifications have been made by Faltings [27], Zannier [28], and Corvaja [29] since the 1980s.
Recall that algebraic numbers are the roots of non-zero polynomial equations with integer coefficients. Another result partially attributed to Siegel, The Thue-Siegel-Roth Theorem, gives absolute bounds on the quality of rational approximations to algebraic numbers [30]. The bounds furnished are not those imposed by limits of computational power. Rather, they are theoretical bounds imposed by the structure of the real numbers that cannot be transcended either by technological innovation or clever use of existing algorithms. Thue-Siegel-Roth states that algebraic numbers of degree greater than 1 cannot be approximated “too closely” by rational numbers. Diophantine equations help researchers understand the distribution and scarcity of these rational approximations to algebraic numbers [30]. It also has applications in other areas of mathematics, such as algebraic geometry, transcendental number theory, and the study of continued fractions [31]. Thue-Siegel-Roth Theorem is a strengthening of the earlier result known as Liouville’s Theorem [32], which states that any algebraic number α of degree d > 1 can only be approximated by rational numbers p/q to within |α – p/q| > C(α) / q^d, where C(α) depends upon α. Thue-Siegel-Roth improves this bound to q^(2-ε), which is significantly stronger for numbers of higher degree [30]. The Bombieri-Lang Conjecture, is a deep conjecture about the distribution of rational points on algebraic varieties that would furnish another general paradigm for looking at Diophantine equations [33]. Bombieri-Lang is an active area of research in computational and pure number theory to which Silicon Valley could contribute computational power and innovative ideas from computer science and simulation.
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