On Galois representations associated to Hilbert modular forms.
On Galois representations defined by torsion points of modular elliptic curves
On Galois structure of the integers in cyclic extensions of local number fields
Let be a rational prime, be a finite extension of the field of -adic numbers, and let be a totally ramified cyclic extension of degree . Restrict the first ramification number of to about half of its possible values, where denotes the absolute ramification index of . Under this loose condition, we explicitly determine the -module structure of the ring of integers of , where denotes the -adic integers and denotes the Galois group Gal. In the process of determining this structure,...
On gaps between numbers with a large prime factor
On gaps between numbers with a large prime factor. II
On gaps in Rényi -expansions of unity for an algebraic number
Let be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi -expansion of unity which controls the set of -integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in are shown to exhibit a “gappiness” asymptotically bounded above by , where is the Mahler measure of . The proof of this result provides in a natural way a new classification of algebraic numbers with classes called Q...
On Garcia numbers.
On Gauss sum characters of finite groups and generalized Bernoulli numbers
On Gauss's proof of Seeber's Theorem.
On Gelfond’s conjecture about the sum of digits of prime numbers
The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.
On general L-functions
On general local reciprocity maps.
On general proof of Fermat's last theorem
On general proof of Fermat's Last Theorem - Epilogue
On generalised arithmetic and geometric progressions
On generalization of continued fraction of Gauss.
On generalizations of the Stirling number triangles.
On generalized bihyperbolic Mersenne numbers
In this paper, a new generalization of Mersenne bihyperbolic numbers is introduced. Some of the properties of presented numbers are given. A general bilinear index-reduction formula for the generalized bihyperbolic Mersenne numbers is obtained. This result implies the Catalan, Cassini, Vajda, d'Ocagne and Halton identities. Moreover, generating function and matrix generators for these numbers are presented.
On generalized elite primes.
On generalized Fermat equations of signature (p,p,3)
This paper focuses on the Diophantine equation , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if , where is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].