Legendre's Conjecture: A Proof via the Tower Sieve with Multi-Sieve Compensation Method
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2026
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| author | zhang, xiang zhang, yue zhang, fadong |
| author_facet | zhang, xiang zhang, yue zhang, fadong |
| contents | <p>Legendre's conjecture asserts that for any positive integer $N$, there is at least one prime in the interval $[N^2, (N+1)^2]$. This paper presents a rigorous proof of this conjecture within the framework of the multi-sieve compensation method. Using the square interval property, the problem is reduced to finding numbers in the interval $A=[0, L-1]$ (where $L = 2N+2$) that are not divisible by any prime $\le N+1$. We construct a large translation interval $U = B \cup C$, where $B = [0, mQ_t-1]$ ($m = P_{t+1}$) consists of complete residue systems modulo $Q_t$, and $C = [mQ_t, mQ_t+L-1]$ is translation equivalent to $A$. Applying the multi-sieve compensation method on $U$, leveraging the huge length of $U$ to ensure complete periods at every step, we prove $N(A) > \frac{2N+2}{6} \prod_{i=3}^t (1-2/P_i)$. Using explicit lower bounds from Mertens' theorem, we show $N(A) > 0$ for $N \ge 10^6$, and $N(A) \to \infty$. For $N < 10^6$, the conjecture can be verified directly. This completes the proof of Legendre's conjecture.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20373198 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Legendre's Conjecture: A Proof via the Tower Sieve with Multi-Sieve Compensation Method zhang, xiang zhang, yue zhang, fadong <p>Legendre's conjecture asserts that for any positive integer $N$, there is at least one prime in the interval $[N^2, (N+1)^2]$. This paper presents a rigorous proof of this conjecture within the framework of the multi-sieve compensation method. Using the square interval property, the problem is reduced to finding numbers in the interval $A=[0, L-1]$ (where $L = 2N+2$) that are not divisible by any prime $\le N+1$. We construct a large translation interval $U = B \cup C$, where $B = [0, mQ_t-1]$ ($m = P_{t+1}$) consists of complete residue systems modulo $Q_t$, and $C = [mQ_t, mQ_t+L-1]$ is translation equivalent to $A$. Applying the multi-sieve compensation method on $U$, leveraging the huge length of $U$ to ensure complete periods at every step, we prove $N(A) > \frac{2N+2}{6} \prod_{i=3}^t (1-2/P_i)$. Using explicit lower bounds from Mertens' theorem, we show $N(A) > 0$ for $N \ge 10^6$, and $N(A) \to \infty$. For $N < 10^6$, the conjecture can be verified directly. This completes the proof of Legendre's conjecture.</p> |
| title | Legendre's Conjecture: A Proof via the Tower Sieve with Multi-Sieve Compensation Method |
| url | https://doi.org/10.5281/zenodo.20373198 |