8 research outputs found
Lucas non-Wieferich primes in arithmetic progressions and the conjecture
In this paper, we improve the lower bound for the number of Lucas
non-Wieferich primes in arithmetic progressions. More precisely, for any given
integer there are Lucas non-Wieferich primes ,
under the assumption of the conjecture for number fields.Comment: 9 pages, some revisions done, result unchange
Congruences for Wolstenholme primes
A prime number is said to be a Wolstenholme prime if it satisfies the
congruence . For such a prime
, we establish the expression for
given in terms of the sums (.
Further, the expression in this congruence is reduced in terms of the sums
(). Using this congruence, we prove that for any Wolstenholme
prime, Moreover, using a recent result of the author \cite{Me}, we prove that the
above congruence implies that a prime necessarily must be a Wolstenholme
prime.
Applying a technique of Helou and Terjanian \cite{HT}, the above congruence
is given as the expression involving the Bernoulli numbers.Comment: pages 1
-adic valuation of harmonic sums and their connections with Wolstenholme primes
We explore a conjecture posed by Eswarathasan and Levine on the distribution
of -adic valuations of harmonic numbers that states
that the set of the positive integers such that divides the
numerator of is finite. We proved two results, using a
modular-arithmetic approach, one for non-Wolstenholme primes and the other for
Wolstenholme primes, on an anomalous asymptotic behaviour of the -adic
valuation of when the -adic valuation of equals exactly 3
On Wolstenholme's theorem and its converse
AbstractFor any positive integer n, let wn=(2nβ1nβ1)=12(2nn). Wolstenholme proved that if p is a prime β©Ύ5, then wpβ‘1(modp3). The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers wn, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers