The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.

In this study, we find all Narayana numbers which are expressible as sums of two base b repdigits. The proof of the main result uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker–Davenport reduction method.

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where a = 0, b are complex numbers, have studied in [16]. In this paper, we show that Bernoulli polynomials Bp(x) can be written in terms of the numbers S 1,x (p, k), and then use the known results for S 1,x (p, k) to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.

The aim of this article is to investigate two new classes of quaternions, namely, balancing and Lucas-balancing quaternions that are based on balancing and Lucas-balancing numbers, respectively. Further, some identities including Binet’s formulas, summation formulas, Catalan’s identity, etc. concerning these quaternions are also established.