On Primes of the Form 2x-q (where q is a prime less than or equal to x) and the Product of the Distinct Prime Divisors of an Integer (Revised): A Function Approach to Proving the Goldbach Conjecture by Mathematical Induction

This paper is really an attempt to solve the age-old problem of the Goldbach Conjecture, by restating it in terms of primes of the form 2x-q (where q is a prime less than or equal to x). Restating the problem merely requires us to ask the question: Does a prime of form 2x-q lie in the interval [x, 2x]? We begin by introducing the product, m, of numbers of the form 2x-q. Using the geometric series, an upper bound is estimated for the function m. Next, we prove a theorem that states every even number, 2x, that violates Goldbach’s Conjecture must satisfy an inequality involving a simple multiplicative function defined as the product, ρ(m), of the distinct prime divisors of m. A proof of the Goldbach Conjecture is then evident by contradiction as a corollary to the proof of the inequality.

Eulerian polynomials via the Weyl algebra action

Through the action of the Weyl algebra on the geometric series, we establish a generalization of the Worpitzky identity and new recursive formulae for a family of polynomials including the classical Eulerian polynomials. We obtain an extension of the Dobiński formula for the sum of rook numbers of a Young diagram by replacing the geometric series with the exponential series. Also, by replacing the derivative operator with the q-derivative operator, we extend these results to the q-analogue setting including the q-hit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.

A further q-analogue of Van Hamme’s (H.2) supercongruence for primes p ≡ 3(mod4)

Long and Ramakrishna [Some supercongruences occurring in truncated hyper- geometric series, Adv. Math. 290 (2016) 773–808] generalized the (H.2) supercongruence of Van Hamme to the modulus [Formula: see text] case. In this paper, we give a [Formula: see text]-analogue of Long and Ramakrishna’s result for [Formula: see text]. A [Formula: see text]-congruence modulo the fourth power of a cyclotomic polynomial, which is a deeper [Formula: see text]-analogue of the (A.2) supercongruence of Van Hamme for [Formula: see text], is also formulated.

Chasing after cancellations: Revisiting a classic identity that implies the Rogers–Ramanujan identities

In this paper, we revisit an identity which was proven by Ramanujan and from which he deduced the famous identities that are named after him and L. J. Rogers. Unlike Ramanujan’s proof (which uses the method of [Formula: see text]-difference equations), we examine directly the [Formula: see text]-coefficients involved. We isolate and identify terms that cancel each other. Once these terms are paired up and canceled, we only need the geometric series to complete the proof.