Geoffrey B. Campbell

activity

20122025

most citedA manufacturable platform for photonic quantum computing

24 citations · 28 across the 18 of their papers we have counts for

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12 papers · 1 filter

math.NT2024★ 2 cited

VPV Identities related to $x^{y} = y^{x}$ and $x^{y} y^{x} = v^{w} w^{v}$

Geoffrey B. Campbell

We cover rational and integer solutions for the equations $x^{y} = y^{x}$ and $x^{y} y^{x} = v^{w} w^{v}$ . The former equation solutions go back to Euler, and the latter equa…

math.NT2019

Some $n$ -space $q$ -binomial theorem extensions and similar identities

Geoffrey B Campbell

We give an $n$ -space generalized $q$ -binomial theorem, and some new $q$ series identities that resemble the traditional $q$ series partition generating functions. These identities…

math.NT2016

The series that Ramanujan misunderstood

Geoffrey B Campbell, Aleksander Zujev

We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan related to the erroneous theory of distribution of primes by Ramanujan.

math.NT2016

On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n

Geoffrey B Campbell, Aleksander Zujev

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.

math.NT2016

Some equations with features of digit reversal and powers

Geoffrey B. Campbell, Aleksander Zujev

In this paper we consider integers in base 10 like $ab c$ , where $a$ , $b$ , $c$ are digits of the integer, such that $ab c^{2} - (ab c \cdot c ba) = \pm n^{2}$ , where $n$ is a positiv…

math.NT2015

Gaussian integer solutions for the fifth power taxicab number problem

Geoffrey B Campbell, Aleksander Zujev

The famous open problem of finding positive integer solutions to $a^{5} + b^{5} = c^{5} + d^{5}$ is considered, and related solutions are found in two distinct settings: firstly, where $a$ …

VPV Identities related to xy=yx and xyyx=vwwv

Some n-space q-binomial theorem extensions and similar identities

The series that Ramanujan misunderstood

On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n

Some equations with features of digit reversal and powers

Gaussian integer solutions for the fifth power taxicab number problem

VPV Identities related to $x^{y} = y^{x}$ and $x^{y} y^{x} = v^{w} w^{v}$

Some $n$ -space $q$ -binomial theorem extensions and similar identities