A Rigorous Proof of Legendre's Conjecture Based on the Modulo-6 Framework 基于模六框架的勒让德猜想严格证明
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2026
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| author | Lⅰu, XingMing |
| author_facet | Lⅰu, XingMing |
| contents | <p>This paper presents a rigorous proof of Legendre's Conjecture based on the modulo-6 classification algebraic framework and the safe residue class sieve. Legendre's Conjecture asserts that for every positive integer n, there exists at least one prime between n² and (n+1)². The proof proceeds as follows: the problem of the existence of a prime in the interval is transformed into the distribution problem of safe residue classes under the modulo-6 framework. By selecting the truncation set K to consist of all primitive indices whose first terms do not exceed (n+1)², the Chinese Remainder Theorem and the Periodic Decomposition Theorem guarantee that the safe residue classes are strictly uniformly distributed within a modulo-L cycle. An exact expression for the number of safe residue classes in the interval is obtained using the group-theoretic counting formula. By invoking the spectral positivity theorem for safe residue classes, it is rigorously proved that the fluctuation of the number of safe residue classes in intervals of equal length is bounded by a constant depending only on K. It follows that when the expected growth value over the interval exceeds this constant, the number of safe residue classes in the interval is strictly greater than zero, and consequently there exists at least one prime in the interval. The entire proof remains within an elementary algebraic framework.</p> <p>本文在模六分类代数框架与安全剩余类筛法的基础上,给出勒让德猜想的严格证明。勒让德猜想断言,对任意正整数n,在n²到(n+1)²之间至少存在一个素数。证明思路如下:将区间上的素数存在性问题转化为模六框架下安全剩余类的分布问题。选取截断集K为全体满足首项不超过(n+1)²的本原指标,由中国剩余定理和周期分解定理,安全剩余类在模L周期内严格均匀分布。利用群论计数公式得到区间内安全剩余类个数的精确表达式。通过引入安全剩余类谱正定性定理,严格证明等长区间内安全剩余类个数的波动受限于一个仅依赖于K的常数。由此推出当区间长度的期望增长值大于该常数时,区间内安全剩余类个数严格大于零,进而推出区间内至少存在一个素数。整个证明保持在初等代数框架内。</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19859842 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | A Rigorous Proof of Legendre's Conjecture Based on the Modulo-6 Framework 基于模六框架的勒让德猜想严格证明 Lⅰu, XingMing Legendre's conjecture 勒让德猜想 Modulo-6 framework 模六框架 Safe residue class sieve 安全剩余类筛法 Periodic decomposition theorem 周期分解定理 Spectral positivity 谱正定性 <p>This paper presents a rigorous proof of Legendre's Conjecture based on the modulo-6 classification algebraic framework and the safe residue class sieve. Legendre's Conjecture asserts that for every positive integer n, there exists at least one prime between n² and (n+1)². The proof proceeds as follows: the problem of the existence of a prime in the interval is transformed into the distribution problem of safe residue classes under the modulo-6 framework. By selecting the truncation set K to consist of all primitive indices whose first terms do not exceed (n+1)², the Chinese Remainder Theorem and the Periodic Decomposition Theorem guarantee that the safe residue classes are strictly uniformly distributed within a modulo-L cycle. An exact expression for the number of safe residue classes in the interval is obtained using the group-theoretic counting formula. By invoking the spectral positivity theorem for safe residue classes, it is rigorously proved that the fluctuation of the number of safe residue classes in intervals of equal length is bounded by a constant depending only on K. It follows that when the expected growth value over the interval exceeds this constant, the number of safe residue classes in the interval is strictly greater than zero, and consequently there exists at least one prime in the interval. The entire proof remains within an elementary algebraic framework.</p> <p>本文在模六分类代数框架与安全剩余类筛法的基础上,给出勒让德猜想的严格证明。勒让德猜想断言,对任意正整数n,在n²到(n+1)²之间至少存在一个素数。证明思路如下:将区间上的素数存在性问题转化为模六框架下安全剩余类的分布问题。选取截断集K为全体满足首项不超过(n+1)²的本原指标,由中国剩余定理和周期分解定理,安全剩余类在模L周期内严格均匀分布。利用群论计数公式得到区间内安全剩余类个数的精确表达式。通过引入安全剩余类谱正定性定理,严格证明等长区间内安全剩余类个数的波动受限于一个仅依赖于K的常数。由此推出当区间长度的期望增长值大于该常数时,区间内安全剩余类个数严格大于零,进而推出区间内至少存在一个素数。整个证明保持在初等代数框架内。</p> |
| title | A Rigorous Proof of Legendre's Conjecture Based on the Modulo-6 Framework 基于模六框架的勒让德猜想严格证明 |
| topic | Legendre's conjecture 勒让德猜想 Modulo-6 framework 模六框架 Safe residue class sieve 安全剩余类筛法 Periodic decomposition theorem 周期分解定理 Spectral positivity 谱正定性 |
| url | https://doi.org/10.5281/zenodo.19859842 |