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A primality criterion based on a Lucas' congruence
Let be a prime. In 1878 \'{E}. Lucas proved that the congruence holds for any nonnegative integer
. The converse statement was given in Problem 1494 of
{\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In
this note we generalize this converse assertion by the following result: If
and are integers such that for every integer , then is a prime and
is a power of .Comment: 6 page
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
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
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