{"id":"hexagon/2610.00504","pdf":"https://hexagonmath.org/pdf/2610.00504v1","url":"https://hexagonmath.org/abs/2610.00504v1","tags":["math.GM","msc:11A41","Legendre's Conjecture","Prime Numbers"],"score":0,"title":"Minimum Prime Bound in Square Intervals","domain":"math.NT","source":"hexagon","authors":["Min Min Oo Oo"],"reviews":0,"subject":"math.NT","summary":"This paper presents a proof of Legendre’s Conjecture by analyzing the distribution of odd composite and their elimination through odd composite sequences. The odd integer sequence is expressed as 𝑂≥3 = 3 + 2𝑟𝑂, while odd composite numbers are represented as 𝑂𝑝 = 𝑝2 + 2𝑝𝑟𝑝, where 𝑝 is prime and 𝑟𝑝 ≥ 0. Each prime-square sequence introduces a systematic elimination of odd composite numbers, reducing the density of primes in the sequence. By sequentially applying elimination factors (𝑞 − 1)/𝑞 for primes 𝑞 ≥ 3, we derive a lower bound for prime density: d/(dr_O ) N_p≥2/p Considering the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}