Volume 12, Issue 5

Abstract. We solve the game of Babylon when played with chips of two colors, giving a winning strategy for the second player in all previously unsolved cases.

Abstract. The study of the structure of infinite words having bounded abelian complexity was initiated by G. Richomme, K. Saari, and L. Q. Zamboni. In this note we define bounded additive complexity for infinite words over a finite subset of . We provide an alternative proof of one of the results of the aforementioned authors.

Abstract. We show that every infinite word on a finite subset of must contain arbitrarily large factors which are “close” to being additive squares . We also show that for all , must contain a factor where all have the same average.

Abstract. The Erdős–Borwein constant is a sum over all Mersenne reciprocals, namely Although E is known to be irrational, the nature of the asymptotic distribution of its binary strings remains mysterious. Herein, we invoke nontrivial properties of the classical divisor function to establish that the googol-th bit, i.e., the bit in position to the right of the point is a `1.' Equivalently, the integer is odd. Methods are also indicated for resolving contiguously the first bits (one full Terabyte) in about one day on modest machinery.

Abstract. We demonstrate, by considering each base in the range 2 through 9, that no odd repdigit with a base in that range is a perfect number.

Abstract. Recently, Gu, Lai and Liang proved necessary and sufficient conditions for a given sequence of positive integers to be the degree sequence of a line-Hamiltonian multigraph. Our goal in this note is to utilize this result to prove a closed formula for the function , the number of degree sequences with degree sum representable by line-Hamiltonian multigraphs. Indeed, we give a truly elementary proof that where is the number of unrestricted integer partitions of j .

Abstract. If E is a subset of , then the n -th characteristic ideal of the algebra of rational polynomials taking integer values on E , , is the fractional ideal consisting of 0 and the leading coefficients of elements of of degree no more than n . For p a prime the characteristic sequence of is the sequence of negatives of the p -adic values of these ideals. We give recursive formulas for these sequences for the sets by describing how to recursively p -order them in the sense of Bhargava. We describe the asymptotic behavior of these sequences and identify primes, p , and exponents, d , for which there is a formula in closed form for the terms.

Abstract. For all positive integers s and t , Brown et al. defined to be the smallest positive integer N such that every 2-coloring of has a monochromatic homothetic copy of {1,1+ s ,1+ s + t }. They proved that for all  s ,  t and that equality holds in the case where both and with and in many other cases. They also proved that for all positive integers  m , or . In this paper, we show that and that for all the other ( s , t ), .

Abstract. We obtain several versions of the sum-product theorem with k -fold sums and product sets. We also give several applications of these estimates.

Abstract. An infinite word x over an alphabet with b letters has full complexity if for each all the b m words of length m are factors of x . We prove that the periods of in the 2-adic expansion approach full complexity for increasing k : For any , the periods for have complexity . Amazingly, these words occur in the period almost the same number of times. On the way, first we prove the same for the binary period. We get a similar result for the decimal period of 1/3 k .

Abstract. A new proof is given for the k -fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called k -in- n dissections.

Abstract. A More Sums Than Differences (MSTD) set is a set of integers whose sumset is larger than its difference set . While it is known that as a positive percentage of subsets of are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman (2010) gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of is at least for some , significantly larger than the previous constructions, which were on the order of . We generalize their method and explicitly construct a large family of sets A with . The additional sums and differences allow us greater freedom than by Miller, Orosz and Scheinerman (2010), and we find that for any the density of such sets is at least . In the course of constructing such sets we find that for any integer k there is an A such that , and show that the minimum span of such a set is 30.

Abstract. The proof of the existence of infinitely many Carmichael numbers depends heavily upon the study of the Carmichael lambda function . In this paper, we study which types of forms this quantity can and cannot take. In particular, for a Carmichael number m , we prove that this can never be of the form . Moreover, we prove that if this is of the form , then either or 127 and m is one of just eight possible values or else m is divisible by a Fermat prime that is not currently among the known Fermat primes.

Abstract. We explore a variation of the isoperimetric problem in which integer subsets take the role of geometric figures. More specifically, we consider the sequence introduced by Miller, Morgan, Newkirk, Pedersen and Seferis (2011) and described in OEIS A186053. We provide the first exact formulas for including recursive relations via auxiliary functions as well as concise and satisfying representations and even quasi-explicit formulas. We also discuss some of the intricate fractal-like symmetry of the sequence and the development of algorithms for computing . We conclude with open questions for further research.

Abstract. Let be an arithmetic function of n variables, where . We study the mean-value of f that is defined to be if this limit exists. We first generalize the Wintner theorem and then consider the multiplicative case by expressing the mean-value as an infinite product over all prime numbers. In addition, we study the mean-value of a function of the form , where g is a multiplicative function of one variable, and express the mean-value by the Riemann zeta function.

Abstract. The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope and respectively. We study extensions to this game where a player may also remove simultaneously tokens from either of the piles and from the other, where are given positive integers and where t ranges over the positive integers. We prove that for certain pairs ( p , q ) the P-positions are identical to those of Wythoff Nim, but for they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs ( p , q ) for which Wythoff Nim's beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim.

Abstract. Wythoff Queens is a classical combinatorial game related to very interesting mathematical results. An amazing one is the fact that the -positions are given by and where . In this paper, we analyze a different version where one player (Left) plays with a chess bishop and the other (Right) plays with a chess knight. The new game (call it Chessfights ) lacks a Beatty sequence structure in the -positions as in Wythoff Queens . However, it is possible to formulate and prove some general results of a general recursive law which is a particular case of a Partizan Subtraction game.

Abstract. Let be subsets of the finite set with and for all , . The ( r 1 ,...,r p )-Stirling number of the second kind, introduced in this paper and denoted by , counts the number of partitions of the set into k classes (or blocks) such that the elements in each , , are in different classes (or blocks). Combinatorial and algebraic properties of these numbers are explored.

Abstract. The number of frequencies of factors of length in a recurrent aperiodic infinite word does not exceed , where is the first difference of factor complexity, as shown by Boshernitzan. Pelantová together with the author derived a better upper bound for infinite words whose language is closed under reversal. In this paper, we further diminish the upper bound for uniformly recurrent infinite words whose language is invariant under all elements of a finite group of symmetries and we prove the optimality of the obtained upper bound.

Abstract. Lehmer's totient problem consists of determining the set of positive integers n such that where is Euler's totient function. In this paper we introduce the concept of k -Lehmer number. A k -Lehmer number is a composite number such that . The relation between k -Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k -Lehmer numbers.