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erdős #855 · second Hardy-Littlewood conjecture

← #854 · #856 → (packet.json; erdosproblems.com)

If $π (x)$ counts the number of primes in $[1, x]$ then is it true that (for large $x$ and $y$ ) $π (x + y) \leq π (x) + π (y)?$

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

This inequality is *exactly* what’s usually called the **second Hardy–Littlewood conjecture**:

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_855 : answer(sorry) ↔
    ∀ᶠ x in atTop, ∀ᶠ y in atTop, π (x + y) ≤ π x + π y

formal-conjectures/855.lean ↗

oeis

A023193 — a(n) gives the largest number k for which there is at least one admissible k-tuple taken from [0, 1, ..., n-1] if the tuple starts with 0. Admissibility is defined in a comment.1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,8,8,9,9,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,12,13,13,14

links

second Hardy-Littlewood conjecture · reference

Hardy-Littlewood prime tuples conjecture · reference

#1204We call a sequence of integers

0 \leq a_{1} < \dots < a_{k}

admissible if it is missing at least one congruence class modulo every prime

p

A (k) = min a_{k}

A (k)

- in particular, is it true that

A (k) \sim k lo g k ?

B (k) = min \frac{a _{1} + \dots + a _{k}}{k} .

status

open

finding.noted · reviewer:will-blair · 1 day

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