Volume 12, Issue 4

Abstract. In this article, we consider increasing sequences of positive integers defined in the following manner. Let the initial terms a 1 and a 2 be given, and for any define to be the smallest integer greater than which cannot be written as a sum of (distinct) previous terms of the sequence. For various parametrized choices of the initial terms, we determine precisely the terms of the sequences obtained by this method. We also conjecture that for all choices of the initial terms, even in a more general setting, the terms of sequences defined in this manner have interesting patterns.

Abstract. This article provides the first bijective proof for a previously “uncounted” Fibonacci number identity of Vajda. Bijections on similar sets that illustrate a related family of Fibonacci number identities are also considered.

Abstract. A length k Newman polynomial is any polynomial of the form (where ). Some Newman polynomials are reducible over the rationals, and some are not. Some Newman polynomials have roots on the unit circle, and some do not. Defining, in a natural way, what we mean by the “proportion” of length  k Newman polynomials with a given property, we prove that   of length 3 Newman polynomials are reducible over the rationals,   of length 3 Newman polynomials have roots on the unit circle,   of length 4 Newman polynomials are reducible over the rationals,   of length 4 Newman polynomials have roots on the unit circle. We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is .

Abstract. We determine the product of the invertible quadratic residues in for odd n . This is a variation on Gauss' generalization of Wilson's theorem. From this we deduce that for twin primes , the product of the invertible quadratic residues in is , where the sign depends on the residue class of p modulo 4. We examine necessary and sufficient conditions for consecutive odd natural numbers to satisfy this property of twin primes. The paper concludes with two open questions.

Abstract. We consider the number of permutations with prescribed up-down structure as a function of two arguments: the number n of elements and the introduced up-down index k of a permutation. We consider sets of permutations for which k is a fixed number and when k is a function of  n . In the first case the number of permutations is a polynomial in  n , the degree of which is defined by  k .

Abstract. We consider a -expansion which makes use of the structure of the corresponding Sturmian sequences, and study some basic properties.

Abstract. In this paper we study the number of carries occurring while performing an addition modulo . For a fixed modular integer t , it is natural to expect the number of carries occurring when adding a random modular integer a to be roughly the Hamming weight of t . Here we are interested in the number of modular integers in producing strictly more than this number of carries when added to a fixed modular integer . In particular it is conjectured that less than half of them do so. An equivalent conjecture was proposed by Tu and Deng in a different context. Although quite innocent, this conjecture has resisted different attempts of proof and only a few cases have been proved so far. The most manageable cases involve modular integers t whose bits equal to are sparse. In this paper we continue to investigate the properties of , the fraction of modular integers a to enumerate, for t in this class of integers. Doing so we prove that has a polynomial expression and describe a closed form for this expression. This is of particular interest for computing the function giving and studying it analytically. Finally, we bring to light additional properties of in an asymptotic setting and give closed-form expressions for its asymptotic values.

Abstract. We use a probabilistic method to evaluate the limit of , where is the Euler totient function and r is a nonnegative integer. We extend the probabilistic method to evaluate two other generalized types of series that involve Euler's totient function. In addition to the probabilistic method, an analytic approach is presented to evaluate the series when the exponent parameter r is a positive real number.

Abstract. In 1849, Alphonse de Polignac conjectured that every odd positive integer can be written in the form , for some integer and some prime p . In 1950, Erdős constructed infinitely many counterexamples to Polignac's conjecture. In this article, we show that there exist infinitely many positive integers that cannot be written in either of the forms or , where is a Fibonacci number, and p is a prime.

Abstract. In this paper, we extend the definition of the Nathanson height from points in projective spaces over to points in projective spaces over arbitrary finite fields. If , then the Nathanson height is where with the field norm and the element of congruent to modulo p . We investigate the basic properties of this extended height, provide some bounds, study its image on the projective line and propose some questions for further research.

Abstract. Let be a sequence of elements from an additive finite abelian group G , and let denote the cross number of S . A zero-sum sequence S of nonzero elements from G is called a unique factorization sequence if S can be written in the form uniquely, where all are minimal zero-sum subsequences of S . In this short note we investigate the following invariant of G concerning the cross number of unique factorization sequences. Define We determine for some special groups including the cyclic groups of prime power order.

Abstract. We find an explicit bound, in terms of g when it is even, for the largest element of an aliquot cycle of repdigits to base g .

Abstract. Let be the largest n such that for every set with at most n elements there exists at least one element in with less than K representations. We show a new lower bound for :

Abstract. A sufficient condition for a monic integer polynomial to be a semi-CNS polynomial is presented. This result infers a unified proof of two well-known theorems on irreducible CNS polynomials thereby extending them to reducible polynomials.

Abstract. We solve the currently smallest open case in the 1976 problem of Molluzzo on , namely the case . This amounts to constructing, for all positive integers n congruent to 0 or 7 mod 8, a sequence of integers modulo 4 of length n generating, by Pascal's rule, a Steinhaus triangle containing 0, 1, 2, 3 with equal multiplicities.

Abstract. Consider a multiplicative function taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that for all primes p in some finite set  P , then the answer is yes. We also discuss a result of Bronšteĭn that shows that functions modified from characters at a finite number of places must have unbounded partial sums.

Abstract. A partition of the set is a collection of nonempty pairwise disjoint subsets of (called blocks ) whose union equals . In this paper, we find exact formulas and/or generating functions for the number of partitions of with k blocks, where k is fixed, which avoid 3-letter patterns of type or , providing generalizations in several instances. In the particular cases of , , and , we are only able to find recurrences and functional equations satisfied by the generating function, since in these cases there does not appear to be a simple explicit formula for it.