Volume 12, Issue 3

Abstract. Let p , with or without subscripts, always denote a prime number. In this paper we are able to establish two localized results on a theorem of Prachar which states that almost all positive even integers n can be written as . As a consequence of one result, we prove additionally that each sufficiently large odd integer N can be represented as with for .

Abstract. Using a clear and straightforward approach, we discover and prove new binary digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. Numerous experimentally discovered and hitherto unproved binary BBP-type formulas are also proved.

Abstract. In this paper, we solve a general, four-parameter recurrence by both algebraic and combinatorial methods. The Bell numbers and some closely related sequences are solutions to the recurrence corresponding to particular choices of the parameters.

Abstract. Let A be the set of nonzero k -th powers in and denote the minimal n such that . We use sum-product estimates for and , following the method of Glibichuk and Konyagin to estimate . In particular, we obtain that for provided that exists.

Abstract. There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analogous question counting the number of k -tuples of nonnegative integers none of which is more than of the sum of all the integers. We give an explicit function for the generating function which counts these k -tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question.

Abstract. The van der Waerden number is the smallest integer m such that in every partition of the set there is always a block that contains an arithmetic progression of length . In this paper, we report the exact value of the previously unknown van der Waerden number , some lower bounds of and polynomial upper-bound conjectures for  and . We also present an efficient SAT-encoding of the number for using which we have computed the exact value of and some lower bounds of .

Abstract. Two new versions of the so-called Maker–Breaker Positional Games are defined by József Beck. He defines two players, Picker and Chooser. In each round, Picker takes a pair of elements not already selected and Chooser keeps one and returns the other to Picker. In the Picker–Chooser version Picker plays as Maker and Chooser plays as Breaker, while the roles are swapped in the Chooser–Picker version. The outcome of these games is sometimes very similar to that of the traditional Maker–Breaker games. Here we show that both Picker–Chooser and Chooser–Picker games are NP-hard, which gives support to the paradigm that the games behave similarly while being quite different in definition. We also investigate the pairing strategies for Maker–Breaker games, and apply these results to the game called “Snaky”.

Abstract. The partition function , which denotes the number of partitions of a positive integer n into distinct parts, has been the subject of a dozen papers. In this paper, we study this kind of partition with the additional constraint that the parts are bounded by a fixed integer. We denote the number of partitions of an integer n into distinct parts, each , by . We find a sharp upper bound for , and more, an infinite series lower bound for the partition function . In the last section, we exhibit a group of interesting identities involving that arise from a combinatorial problem.

Abstract. The Stern sequence (also called Stern's diatomic sequence ) is defined by the recurrence relations , , and in general by and . In this note we prove a new identity for Stern's sequence. In particular, we show that if e and a are nonnegative integers, then for any integer r with , we have It seems that this is the first correlation-type identity concerning Stern's sequence in the literature.

Abstract. Let be a sequence of integers. A primitive prime divisor of a term is a prime which divides but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial is a sequence of integers where the n -th term is the n -th iterate of at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that for in , where and , every iterate in the zero orbit of contains a primitive prime divisor whenever zero has an infinite orbit. If , then every iterate after the first contains a primitive prime divisor.