On the first case of Fermat's last theorem

Let p be an odd prime and suppose that for some a, b, c ? $\text{Z}$\p$\text{Z}$ we have that a^p + b^p + c^p = 0. In Part I a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given. As is known, these congruences have Wieferich and Mirimanoff criteria (2^{p ? 1} ≡ 1 mod p² and 3^{p ? 1} ≡ 1 mod p²) as immediate consequences. Mirimanoff's congruences are expressed in the form of polynomial congruences $\text{P}{}_{\mathrm{m}}\text{(}\text{?a}\text{b}\text{) ≡ 0}\text{mod}\text{p}$, 1 ≤ m ≤ p ? 1, and these polynomials P_m(X) are characterized by means of simple relations. In Part II a complement to Kummer-Mirimanoff congruences is given under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field.     

Remarks concerning possible connections between Fermat's last theorem and integral p×p circulants

Let p be a prime and x, y integers. Then x^p+y^p occurs as the determinant of an integral p×p circulant. Some problems concerning the set of values assumed by det C as C runs through the set of integral p×p circulants are given. The problem of finding sufficient conditions on an integral p×p circulant C which ensure that C can be factored in the form C₁C'₁ (where ' denotes transpose) with C₁ an integral circulant is referred to, and it is conjectured that such a factorization is possible if C is unimodular and positive definite symmetric and $\text{1}\text{2}$(p–1) is also a prime.     

On Petersen's graph theorem

In this paper we prove the following: let G be a graph with e_G edges, which is (k ? 1)-edge- connected, and with all valences ?k. Let 1?r?k be an integer, then G contains a spanning subgraph H, so that all valences in H are ?r, with no more than ?re_G?k? edges. The proof is based on a useful extension of Tutte's factor theorem [4,5], due to Lovász [3]. For other extensions of Petersen's theorem, see [6,7,8].     

On Dilworth's decomposition theorem

For every countable po set P without infinite chains there exists a partition (C_j: j?I) of P into pairwise disjoint chains and an antichain A such that C_j∩A≠Ø for every i?I.     

Adams theorem on Bernoulli numbers revisited

If we denote B_n to be nth Bernoulli number, then the classical result of Adams (J. Reine Angew. Math. 85 (1878) 269) says that p^?|n and (p−1)?n, then p^?|B_n where p is any odd prime p>3. We conjecture that if (p−1)?n, p^?|n and p?+1?n for any odd prime p>3, then the exact power of p dividing B_n is either ? or ?+1. The main purpose of this article is to prove that this conjecture is equivalent to two other unproven hypotheses involving Bernoulli numbers and to provide a positive answer to this conjecture for infinitely many n.     

On the converse of singer's theorem

A construction is given in which the nonzero elements of a planar difference set give rise to a totally symmetric quasi-group. Examples are provided which suggest that the quasi-group is essentially the additive group of the field. The evidence supports the conjecture that the converse of Singer's theorem holds. The Multiplier Theorem is used to characterize when the totally symmetric quasi-groups are totally symmetric loops. The results extend to Abelian group difference sets (λ = 1).     

On Schmerl's effective version of Brooks' theorem

A new proof is given of Schmerl's recent result that a highly recursive graph G with χ(G) ≤ k according to Brooks' theorem, has a recursive k-colouring.     

On Bayes' theorem and the inverse Bernoulli theorem

A recent assertion by S. M. Stigler that Thomas Bayes was perhaps anticipated in the discovery of the result that today bears his name is exposed to further scrutiny here. The distinction between Bayes' theorem and the inverse Bernoulli theorem is examined, and pertinent early writings on this matter are discussed. A careful examination of the difference between these two theorems leads to the conclusion that a result given by David Hartley in 1749 is more in line with the inverse Bernoulli theorem than with Bayes' result, and it is suggested that there is not sufficient evidence to remove Bayes from his place as originator of the method adopted.     

On the extension of von Neumann-Aumann's theorem

We give a measurable selection theorem which generalizes von Neumann-Aumann's theorem when the domain of definition is an abstract measurable space and the range space is a Suslin space.As application we give a measurable implicit function theorem and a parametrized version of Choquet's theorem on integral representation.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.     

On Carlitz's q-Bernoulli numbers

Properties of q-extensions of Bernoulli numbers and polynomials which generalize those satisfied by B_k and B_k(x) are used to construct q-extensions of p-adic measures and define a q-extension of p-adic Dirichlet L-series.     

On Bernoulli and Euler numbers

The main purpose of this paper is to investigate some basic relations (e.g., Voronoi's and Kummer's congruences) of Bernoulli and Euler numbers by manipulating Euler factors in a natural way.     

On Kersting's proof of the central limit theorem

By an adaptation of a method originally invented by G. Kersting [1] for the calculation of the limiting distribution of Markovian processes the central limit theorem (CLT) is proven. Only the case of equal variances is considered.     

On hypergeometric Bernoulli numbers and polynomials

We provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.