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Minimum Prime Bound in Square Intervals

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Hexagonhexagon/2610.00504Oct 23, 2026math.NTscore 0

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 𝑂β‰₯3 = 3 + 2π‘Ÿπ‘‚, while odd composite numbers are represented as 𝑂𝑝 = 𝑝2 + 2π‘π‘Ÿπ‘, where 𝑝 is prime and π‘Ÿπ‘ β‰₯ 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 (π‘ž βˆ’ 1)/π‘ž for primes π‘ž β‰₯ 3, we derive a lower bound for prime density: d/(dr_O ) N_pβ‰₯2/p Considering the…

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