Volume 11, Issue 2

Fix α > 0, and sample N integers uniformly at random from {1, 2, . . . , ⌊ e α N ⌋}. Given η > 0, the probability that the maximum of the pairwise GCDs lies between N 2– η and N 2+ η converges to 1 as N → ∞. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order N 2 /log( N ). The proof generalizes to any arithmetical semigroup where a suitable form of the prime number theorem is valid.

The so-called ζ -analogues of the Stirling numbers of the first and second kind are considered. These numbers cover ordinary binomial and Gaussian coefficients, p, q -Stirling numbers and other combinatorial numbers studied with the help of object selection, Ferrers diagrams and rook theory. Our generalization includes these and now also the p, q -binomial coefficients. This special subfamily of F -nomial coefficients encompasses among others, Fibonomial ones. The recurrence relations with generating functions of the ζ -analogues are delivered here. A few examples of ζ -analogues are presented.

Simson's identity is a well-known Fibonacci identity in which the difference of certain order 2 products has a particularly pleasing form. Other old and beautiful identities of a similar nature are attributed to Catalan, Gelin and Cesàro, and Tagiuri. Catalan's identity can be described as a family of product difference Fibonacci identities of order 2 with 1 parameter. In Section 2 of this paper we present four families of product difference Fibonacci identities that involve higher order products. Being self-dual, each of these families may be regarded as a higher order analogue of Catalan's identity. We also state two conjectures that give the form of similar families of arbitrary order. In the final section we give other interesting product difference Fibonacci identities.

The purpose of the paper is to give some sequences that converge quickly to a generalization of Euler's constant, i.e., the limit of the sequence where a ∈ (0, +∞).

Let ƒ( n ) be the number of unordered factorizations of a positive integer n as a product of factors > 1. In this paper, we show that the number of distinct values of ƒ( n ) below x is at most exp(9(log x 2/3 ) for all x ≥ 1.

Let φ denote Euler's totient function. The frequency with which φ ( n ) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which φ ( n ) is a sum of two squares has been studied by Banks, Luca, Saidak, and Shparlinski. Here we look at the corresponding three-squares question. We show that φ ( n ) is a sum of three squares precisely seven-eighths of the time. We also investigate the analogous problem with φ replaced by Carmichael's λ -function. We prove that the set of n for which λ ( n ) is a sum of three squares has lower density > 0 and upper density < 1.

For sequences defined in terms of 2-adic valuations, we exploit the intrinsic periodic behaviour obtained by a double summation. The tool is the Mellin–Perron formula.

In this paper, we consider the diophantine equation X t + Y t = BZ t where X, Y, Z are nonzero coprime integers. We prove that this equation has no nontrivial solution with the exponent t dividing Z under certain conditions on t and B .

We determine the p -exponent in many of the coefficients of , where is the power series for log(1 + x )/ x and t is any integer. In our proof, we introduce a variant of multinomial coefficients. We also characterize the power series x /log(1 + x ) by certain zero coefficients in its powers.

Recently, Tu and Deng proposed a combinatorial conjecture about binary strings, and, on the assumption that the conjecture is correct, they obtained two classes of Boolean functions which are both algebraic immunity optimal, the first of which are also bent functions. The second class gives balanced functions, which have optimal algebraic degree and the best nonlinearity known up to now. In this paper, by using three different approaches, we prove the conjecture for many cases. We also propose some problems about the weight equations which are related to this conjecture. Because of the scattered distribution, we predict that an exact count is difficult to obtain, in general.

The aim of this paper is to investigate and present the general properties of the (exponential) bipartitional polynomials. After establishing relations between bipartitional polynomials and polynomial sequences of binomial and trinomial type, a number of identities are deduced.

Motivated by the Cauchy–Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result: There is a universal constant ∈ > 0 such that, if is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most three, or the number of pairs of such vertices is at least 1 + ∈ times the number of edges in . We discuss a range of further questions to which this result gives rise.