erdős #5

← #4 · #6 → (packet.json; erdosproblems.com)

Let $C \geq 0$ . Is there an infinite sequence of $n_{i}$ such that $i \to \infty lim \frac{p _{n_{i} + 1} - p _{n_{i}}}{lo g n _{i}} = C ?$

unreviewedOpen. Worked here; no verified result yet.

number theory · open · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (d_n:=p_{n+1}-p_n). Your question is asking whether, for each prescribed (C\ge 0), one can find infinitely many indices (n_i) such that [ \frac{d_{n_i}}{\log n_i}\to C. ] Equivalently, is $C$ a **limit point** of the sequence ({d_n/\log n})?

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llm-hunter · codex 5.2 extra high, gpt 5.2, gpt pro 5.2 · unverified

4 LLM attack(s) recorded (codex 5.2 extra high, gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

OEIS1

A001223 — Prime gaps: differences between consecutive primes.1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,

Connections1

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#852Let

d_{n} = p_{n + 1} - p_{n}

, where

p_{n}

is the

n

th prime. Let

h (x)

be maximal such that for some

n < x

the numbers

d_{n}, d_{n + 1}, \dots, d_{n + h (x) - 1}

are all distinct. Estimate

h (x)

. In particular, is it true that

h (x) > (lo g x)^{c}

for some constant

c > 0

, and

h (x) = o (lo g x)?

A001223 #853Let

d_{n} = p_{n + 1} - p_{n}

, where

p_{n}

is the

n

th prime. Let

r (x)

be the smallest even integer

t

such that

d_{n} = t

has no solutions for

n \leq x

.Is it true that

r (x) \to \infty

? Or even

r (x) / lo g x \to \infty

?A001223

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