18 research outputs found
On the denominators of harmonic numbers
Let be the -th harmonic number and let be its denominator. It
is well known that is even for every integer . In this paper, we
study the properties of . One of our results is: the set of positive
integers such that is divisible by the least common multiple of has density one. In particular, for any
positive integer , the set of positive integers such that is
divisible by has density one.Comment: 6 page
On the p-adic valuation of harmonic numbers
For any prime number p, let Jp be the set of positive integers n such that p divides the numerator of the n-th harmonic
number Hn. An old conjecture of Eswarathasan and Levine states that Jp is finite. We prove that for x ≥ 1 the number of integers in Jp ∩ [1, x] is less than 129p2/3x0.765.
In particular, Jp has asymptotic density zero. Furthermore, we show that there exists a subset Sp of the positive integers, with logarithmic density greater than 0.273, and such that for any n ∈ Sp the p-adic valuation of Hn is equal to −logp n
Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
In 1862 Wolstenholme proved that for any prime the numerator of the
fraction written in reduced form is divisible by , and the numerator of
the fraction
written in reduced form is divisible by . The first of the above
congruences, the so called {\it Wolstenholme's theorem}, is a fundamental
congruence in combinatorial number theory. In this article, consisting of 11
sections, we provide a historical survey of Wolstenholme's type congruences and
related problems. Namely, we present and compare several generalizations and
extensions of Wolstenholme's theorem obtained in the last hundred and fifty
years. In particular, we present more than 70 variations and generalizations of
this theorem including congruences for Wolstenholme primes. These congruences
are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13
textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of
Integer Sequences. In this article, some results of these references are cited
as generalizations of certain Wolstenholme's type congruences, but without the
expositions of related congruences. The total number of citations given here is
189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type
congruences (1862-2012) including more than 70 related results and 106
references. This is in fact version 2 of the paper extended with congruences
(12) and (13
p-AdicL-Functions and Sums of Powers
AbstractWe give an explicitp-adic expansion of ∑npj=1, (j, p)=1j−ras a power series inn. The coefficients are values ofp-adicL-functions
-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