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Math-2794-A1 · revision r1 · on Math-2794

OpenAI claim summary for Math-2794

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 011; 2026-09-17).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Prime predecessors with even squarefree factor count
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Prime predecessors with even squarefree factor count

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 011; 2026-09-17), the asserted result is that infinitely many prime predecessors are squarefree with an even number of prime factors, equivalently $\mu(p-1)=1$.

Mathematical statement of the claimed result

Manuscript 1.
If $\Omega(n)$ counts prime factors with multiplicity, the asserted infinitude is \[ |\{p\text{ prime}:p-1\text{ squarefree},\ \Omega(p-1)\equiv0\pmod2\}|=\infty. \] Squarefreeness makes the same parity statement hold when prime factors are counted without multiplicity.

Outline of the argument

For manuscript 1 (family 011; 2026-09-17), the shifted-correlation framework is extended to a long factor that is itself prime and carries a common parity sign at both endpoints. Major-arc endpoint estimates, rough weights, and squarefree sieving then extract prime predecessors with the required sign.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 011; 2026-09-17).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources