Related papers: Wilson's theorem

We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type…

For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial…

We generalize Mertens' product theorem to Chebotarev sets of prime ideals in Galois extensions of number fields. Using work of Rosen, we extend an argument of Williams from cyclotomic extensions to this more general case. Additionally, we…

For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal…

In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not…

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with…

In this paper we go on to discuss about Stanley's theorem in Integer partitions. We give two different versions for the proof of the generalization of Stanley's theorem illustrating different techniques that may be applied to profitably…

A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.

In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian…

A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number…

We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed…

The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with…

We build on work of Kellendonk, Richard, Tiedra de Aldecoa and others to show that the wave operators for Schr\"{o}dinger scattering theory on $\mathbb{R}^n$ generically have a particular form. As a consequence, Levinson's theorem can be…

We define the resultant of two power series with coefficients in the ring of integers of a $p$-adic field. In order to do this, we prove a universal version of the Weierstrass preparation theorem.

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in…

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…

We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.