# Advancing special topics in number theory

An EU-funded project has developed new analytical tools for computing solutions to complex mathematical equations. Developments will enable the resolution of long-standing issues in number theory and provide an important new module to commercially available software.

One of the most astoundingly simple equations, as far as mathematics is concerned, remained open to debate for over 350 years despite being the theorem with the most unpublished false proofs ever. Fermat’s Last Theorem, postulated in the 1600s, states that the equation xn + yn = zn has no non-zero integer solutions for x, y and z if n is an integer greater than 2 (when n=2, the equation is that of the well-known Pythagorean Theorem). Andrew Wiles finally proved the theorem in 1994, opening the way for advances in numerous related fields of mathematics and, in particular, number theory. European researchers sought to extend the work of Wiles and others to the realm of automorphic forms and Hilbert modular forms via the ‘Hilbert modular forms and Diophantine applications’ (HMF) project. Automorphic forms are one of the most complex topics of number theory and have quite broad application, in addition to their use by Wiles to prove Fermat’s Last Theorem. They are intrinsically related to prime numbers, which have recently formed the basis of secure transactions over the Internet. In addition, one of the most important methods for constructing automorphic forms is also foundational to quantum mechanics. Modular forms are a special case of automorphic forms. They are highly symmetric functions on the set of elliptical curves. Hilbert modular forms have numerous applications to so-called Diophantine equations, indeterminate polynomial equations for which the variables can only be integers (including Fermat’s equation). Given that the resolution of many Diophantine problems requires explicit understanding of more difficult Hilbert modular and automorphic forms, HMF researchers developed robust algorithms for computing Hilbert modular forms that were incorporated into the computational algebra software system MAGMA. Project work thus contributed to resolving a key long-standing mathematical conjecture and provided important evidence related to others. HMF project results advanced the field of automorphic forms and contributed important algorithms to a well established commercially available computational software programme. Numerous applications of the work both to basic research in mathematics as well as to applied research regarding number theory can be expected.