Volume 31, 2025, Number 1 (Online First)

In order to investigate the relationship between Gaussian Fibonacci numbers and quantum numbers and to develop both a deeper theoretical understanding in this study, q-Gaussian Fibonacci, q-Gaussian Lucas quaternions and polynomials are taken with quantum integers by bringing a different perspective. Based on these definitions, the Binet formula of these number sequences is found, and some algebraic properties, important theorems, propositions and identities related to the formula are given. Thus, new perspectives are obtained in the analysis and applications of complex systems.

We revisit a classical theorem of Chebyshev about distribution of primes on intervals , and prove a generalization of it. Extending Erdős’ arithmetical-combinatorial argument, we show that for all , there is such that the intervals contain a prime for all . A quantitative lower bound is derived for the number of primes on such intervals. We also give numerical upper bounds for for , and we draw comparisons with existing results in the literature.

This paper investigates a number of congruence properties related to the coefficients of a generalized Fibonacci polynomial. This polynomial was defined to produce properties comparable with those of the standard polynomials of some special functions. Some of these properties are compared with known identities, while others are seemingly characteristic of arbitrary order recurrences. These include generalizations of, and analogies for, results of Appell, Bernoulli, Euler, Hilton, Horadam and Williams. In turn, the theorems lead to conjectures for further development.