Minimum Prime Bound in Square Intervals
Primarily human-written textHuman understanding: all partsmath.NT β Number Theorymath.GM β General Mathematics
Contributed by Min Min Oo Oo β
Version 1 / Oct 23, 2026 / CC BY 4.0
Abstract
This paper presents a proof of Legendreβs Conjecture by analyzing the distribution of odd composite and their elimination through odd composite sequences. The odd integer sequence is expressed as πβ₯3 = 3 + 2ππ, while odd composite numbers are represented as ππ = π2 + 2πππ, where π is prime and ππ 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 β 1)/π for primes π β₯ 3, we derive a lower bound for prime density:
d/(dr_O ) N_pβ₯2/p
Considering the quadratic interval [π2, (π + 1)2], the total change in ππ is Ξππ = (π+1)2βπ2 = π + Β½ . Combining this with the density bound yields a lowest bound of prime count per interval:
Ο_((n^2,(n+1)^2))^LBβ₯β2/p (n+1/2)β=2(pβn+1)
Thus, every quadratic interval [π2, (π + 1)2] contains at least two primes, establishing Legendreβs Conjecture. This elimination-based framework provides a systematic proof based on the distribution of odd integers and odd composite sequences.
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Version history
- v1Submitted by Min Min Oo OoInitial depositCurrentOct 23, 2026