Fermat's Little Theorem via Divisibility of Newton's Binomial

ON MODALITY AND DIVISIBILITY OF POISSON AND BINOMIAL MIXTURES

Some structural aspects of mixtures, in general, have been previously investigated by the author in [I] and [2]. The aim of this article is to investigate some important structural properties of the special cases of Poisson and binomial mixtures in detail. Some necessary and sufficient conditions are arrived at for different modality and divisibility properties of a Poisson mixture based o...

Additions and Corrections "M.H.Alamatsaz- On Modality and Divisibility of Poisson and Binomial Mixtures. J.Sci. I.R. Iran, Vol.l,No.3, Spring 1990

The Power of a Prime That Divides a Generalized Binomial Coefficient

The main idea is to consider generalized binomial coefficients that are formed from an arbitrary sequence C, as shown in (3) below. We will isolate a property of the sequence C that guarantees the existence of a theorem like Kummer’s, relating divisibility by prime powers to carries in addition. A special case of the theorem we shall prove describes the prime power divisibility of Gauss’s gener...

On Some Generalizations of Fermat's, Lucas's and Wilson's Theorems

“Never underestimate a theorem that counts something!” – or so says J. Fraleigh in his classic text [2]. Indeed, in [1] and [4], the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this le...

An Elementary Proof of a Congruence by Skula and Granville

which is an integer. Fermat quotients played an important role in the study of cyclotomic fields and Fermat Last Theorem. More precisely, divisibility of Fermat quotient qp(a) by p has numerous applications which include the Fermat Last Theorem and squarefreeness testing (see [1], [2], [3], [5] and [9]). Ribenboim [10] and Granville [5], besides proving new results, provide a review of known fa...