On Eisenstein's problem
On exactly covering systems of congruences having moduli occurring at most twice
On exceptional systems of random integers.
On exceptions to Szegedy's theorem
On expansions of Gaussian integers with non-negative digits.
On explicit reciprocity laws.
On explicit reciprocity laws. II.
On -thin sets
On factorization of integers with restrictions on the exponents.
On families of nonlinear recurrences related to digits.
On fundamental solutions of binary quadratic form equations
We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation Au²+Buv+Cv²=N, where A>0, N≠0 and B²-4AC is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation u²-dv²=N, where d is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.
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 generalization of continued fraction of Gauss.
On generalized elite primes.
On generalized Landau identities.
On Giuga's Conjecture.
On Grimm's problem relating to factorisation of a block of consecutive integers. II.
On Grosswald's conjecture on primitive roots
Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all and for all .
On happy factorizations.
On Hong’s conjecture for power LCM matrices
A set of distinct positive integers is said to be gcd-closed if for all . Shaofang Hong conjectured in 2002 that for a given positive integer there is a positive integer depending only on , such that if , then the power LCM matrix defined on any gcd-closed set is nonsingular, but for , there exists a gcd-closed set such that the power LCM matrix on is singular. In 1996, Hong proved and noted for all . This paper develops Hong’s method and provides a new idea to calculate...