Survivor gaps
Reproduces the 6,4,2,4,2,4,6,2 gaps of the sieve modulo 30.
Expected: gaps = [6, 4, 2, 4, 2, 4, 6, 2]
View script
idle
Mathematical atlas
PT mathematics / sieve
theorem
Prime gaps and survivor gaps
Reading prime gaps as a limiting case of gaps between sieve survivors.
Plain
The idea
A prime gap is the distance between two neighboring primes. PT starts more simply: a gap is the distance between two positions that survived the sieve.
When the sieve depth reaches the square-root window, the remaining survivors are exactly 1 and the primes. Prime gaps become a survivor-mechanics problem.
circular gaps
Standard
Standard reading
Modulo a primorial $M_A$, residues coprime to $M_A$ form a circular sequence. Successive differences give the sieve gaps.
Exact reduction to primes occurs at $y=\lfloor\sqrt{x}\rfloor$: every composite $n\le x$ has a prime factor $\le\sqrt{x}$ and cannot survive.
Takeaways
- Sieve gaps are exact.
- The square-root window reduces survivors to primes.
- The fine law of prime gaps remains a research frontier.
Technical
Technical formulation
The exact core is $S(x;\lfloor\sqrt{x}\rfloor)=1+\pi(x)-\pi(\lfloor\sqrt{x}\rfloor)$, where $S(x;y)$ counts integers divisible by no prime $\le y$.
The open part is not the reduction to primes, but the fine closed law that would predict each gap $p_{n+1}-p_n$ without already knowing the neighboring primes.
Monograph: ch01_sieve, ch07_convergence, Riemann part.
Formulas
$\varphi(M_A)=M_A\prod_{p\in A}(1-1/p)$
$S(x;\lfloor\sqrt{x}\rfloor)=1+\pi(x)-\pi(\lfloor\sqrt{x}\rfloor)$
public code
Code and scripts
The links below point to public resources or planned GitHub repositories. No local working path is exposed to the reader.