Number Theory

A Symmetrical Strengthening of Legendre's Conjecture Anchored on Even Squares

Authors: Mohammad Aldebbeh

Legendre's Conjecture states that there is a prime number between n2 and (n+1)2 for every positive integer n. This paper proposes a symmetrical, computationally verified strengthening of Legendre's Conjecture by anchoring the search for primes exclusively on even squares. We hypothesize that for every even integer n, there exists a prime q < 2n+1 such that n2+q is prime, and a prime q < 2n-1 such that n2-q is prime. We demonstrate that these two conjectures collectively imply Legendre's Conjecture for all integers. Furthermore, we explore the algebraic links to Polignac's and Goldbach's Conjectures, highlighting how our tight bounding of q bridges these theories. Probabilistic heuristics based on the Prime Number Theorem are provided to justify the logarithmic growth of q. Finally, we present computational evidence verifying both conjectures up to n=108.

Comments: 5 Pages. 2 figures (Note by viXra Admin: Please cite and listed scientific references)

Download: PDF

Submission history

[v1] 2026-03-24 23:33:00

Unique-IP document downloads: 38 times

ai.Vixra.org is a AI assisted e-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. ai.Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.