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Math-2802 · mathematics · Number theory

Many primes with power-smooth predecessors

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed delta in (0,1/4), does the number of primes p in (2x,5x] whose predecessor p-1 has no prime factor exceeding $x^\delta $ grow at least as $x^(1-o(1))$ as x tends to infinity? The error exponent may depend on delta.

Short English statement

For each fixed delta in (0,1/4), does the number of primes p in (2x,5x] whose predecessor p-1 has no prime factor exceeding $x^\delta $ grow at least as $x^(1-o(1))$ as x tends to infinity? The error exponent may depend on delta.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2802-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2802

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; 2026-09-24).

Many primes with power-smooth predecessors

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; 2026-09-24), the asserted result is that infinitely many totient values have more than $n^{1-\varepsilon}$ preimages for every positive $\varepsilon$, using many primes whose predecessors are very smooth.

Mathematical statement of the claimed result

Manuscript 1.
Write $P^+(m)$ for the largest prime factor of a positive integer. For every fixed real $0<\delta<1/4$, the manuscript claims \[ |\{p\text{ prime}:2x<p\le5x,\ P^+(p-1)\le x^{\delta}\}| \ge x^{1-o(1)}\qquad(x\to\infty). \] Equivalently, for each fixed $\delta$ in this range and each $\eta>0$, the count is at least $x^{1-\eta}$ for all sufficiently large real $x$; the threshold can depend on both parameters. The smoothness bound applies to the full shifted integer $p-1$.

Outline of the argument

For manuscript 1 (family 011; 2026-09-24), weighted dilation graphs are transferred to independent residue models, with signed-memory and frequency estimates controlling the error. The resulting supply of smooth prime predecessors yields many distinct prime products but comparatively few totient values.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 011; 2026-09-24).

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

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