Volume 12, Issue 2

Abstract. This article proves a conjecture by S.-C. Liu and J. C.-C. Yeh about Catalan numbers, which states that odd Catalan numbers can take exactly distinct values modulo , namely the values .

Abstract. Nogueira, in 1995, presented a study of the Poincaré map and the related Parry–Daniels map. In this note some variants of this algorithm are presented which seem to have quite different ergodic behavior.

Abstract. We solve a family of quadratic Diophantine equations associated with a simple kind of game. We show that the ternary case, in many ways, is the most interesting and the least arbitrary member of the family.

Abstract. We determine the norm Euclidean orders in a positive definite quaternion algebra over .

Abstract. Winquist's identity plays a vital role in the proof of Ramanujan's congruence . In this paper, we give a new proof of Winquist's identity.

Abstract. The purpose of this work is to extend the theory of finite operator calculus to the multivariate setting, and apply it to the enumeration of certain lattice paths. The lattice paths we consider are ballot paths. A ballot path is a path that stays weakly above the diagonal , starts at the origin, and takes steps from the set . Given a string from the set , we want to count the ballot paths with a given number of occurrences of . In order to use finite operator calculus, we must put some restrictions on the string we wish to keep track of. A ballot path ending on the diagonal can be viewed as a Dyck path, thus all of our results also apply to the enumeration of Dyck paths with a given number of occurrences of . Finally, we give an example of counting ballot paths with a given number of occurrences of two patterns.

Abstract. We shall construct three binary linear codes , , and , respectively associated with the orthogonal groups , , and , with a power of two. Then we obtain recursive formulas for the power moments of Kloosterman and 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via the Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups.

Abstract. We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal's triangle that are divisible by and not divisible by , where is a prime. Using these relations to compute this number can provide significant savings in the number of computational steps.

Abstract. Let the sequence of nonnegative integers be generated by the following conditions. Set the first term , and for all , let be the least integer greater than such that no element of is the average of distinct other elements. Szekeres gave a closed-form description of in 1936, and Layman provided a similar description for in 1999. We first find closed forms for some similar greedy sequences that avoid averages in terms not all the same . Then, we extend the closed-form description of from the known cases when and to any integer . With the help of a computer, we also generalize this to sequences that avoid solutions to specific weighted averages in distinct terms. Finally, from the closed forms of these sequences, we find bounds for their growth rates.

Abstract. Recall that an integer is -free if and only if it is not divisible by for some prime . We give a method to check Robin's inequality, , for -free integers and apply it for . We introduce , a generalization of the Dedekind function defined for any integer , by If is -free, then the sum of divisor function is . We characterize the champions for , as primorial numbers. Define the ratio . We prove that, for all , there exists an integer such that we have for , where . Further, by combinatorial arguments, this can be extended to for all such that . This yields Robin's inequality for . For varying slowly with , we also derive .

Abstract. We investigate 3-adic valuations of generalized harmonic numbers . These valuations are completely determined by the 3-adic expansion of . Moreover, we also give 3-adic valuations of generalized alternating harmonic numbers.