Does there exist a polynomial $f (x) \in Z [x]$ such that all the sums $f (a) + f (b)$ with $a < b$ nonnegative integers are distinct?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=partial) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'partial'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

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

This is a famous open problem in additive number theory.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 11 · open (literature)

theorem erdos_324 : answer(sorry) ↔
    ∃ f : ℤ[X], {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => f.eval (a : ℤ) + f.eval (b : ℤ)

open