ProbXiv
sign in
machine only

Sárközy's Conjecture on Sums and Products Modulo a Prime

Everything below was recorded by a tool. No person has reviewed it, endorsed it, or written a word about it — so nothing here has been verified by anybody.

sarkozy-sums-products-mod-pNumber theoryposed by András Sárközyrecorded: disproved

1 attempt · 1 machine check · no person has looked

Statement

For A⊆FpA \subseteq \mathbb{F}_p let A∗=(A+A)∪(AA)A^* = (A+A) \cup (AA). Sárközy conjectured that for all large primes, every set of size at least cpc\sqrt{p} has A∗=FpA^* = \mathbb{F}_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.

Context

A named Sárközy conjecture in the Gyarmati-Sárközy line of equations over finite fields, an established specialist question with a real literature.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionHarmonic Aristotle with Quanyu Tang ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    Harmonic Aristotle
    people
    Quanyu Tang

    Aristotle produced formal Lean proofs of all four principal statements of the paper (the formalization is public), and "was also used to assist in the preparation of this paper." The counterexample construction itself builds on a classical projective-geometric orbit.

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: partially checked

      Recorded from Lean ·

      scope Lean formalization of the core argument; statement correspondence not independently audited

      All four principal statements formalized and checked in Lean; Wouter van Doorn assisted with the formalization. No independent expert review yet. Tier: the formalization was produced within the project (Aristotle, with van Doorn assisting); no independent statement audit.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.