erdős #218
Let . The set of such that has density , and similarly for . Furthermore, there are infinitely many such that .
Worked, still open.
number theory · open · formalized (Lean) · 0 attempts
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vela registry pull vfr_37aec80d874a0239vela reproduce examples/erdos-problemsevidence
unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
What you wrote is *exactly* **Erdős Problem #218** on consecutive prime gaps, and it is currently listed as **open**. ([Erdős Problems][1])
candidate solution ↗llm-hunter · gpt pro 5.2 · unverified
1 LLM attack(s) recorded (gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 11 · open (literature)
theorem erdos_218.variants.le : {n | primeGap n ≤ primeGap (n + 1)}.HasDensity <| 1 / 2formal-conjectures/218.lean ↗oeis
A064113 — Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.2,15,36,39,46,54,55,73,102,107,110,118,129,160,164,184,187,194,199,218,239,271,272,291,339,358,387,419,426,464,465,508,5A333230 — Positions of weak ascents in the sequence of differences between primes.1,2,3,5,7,8,10,13,14,15,17,20,22,23,26,28,29,31,33,35,36,38,39,41,43,45,46,49,50,52,54,55,57,60,61,64,65,67,69,70,71,73,A333231 — Positions of weak descents in the sequence of differences between primes.2,4,6,9,11,12,15,16,18,19,21,24,25,27,30,32,34,36,37,39,40,42,44,46,47,48,51,53,54,55,56,58,59,62,63,66,68,72,73,74,77,8
status
open