Erdős Problem #1074

The question

erdos_1074.parts.i. Let $S$ be the set of all $m\ge 1$ such that there exists a prime $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$. Does $$\lim \frac{|S\cap [1,x]|}{x}$$ exist?

erdos_1074.parts.ii. Let $S$ be the set of all $m\ge 1$ such that there exists a prime $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$. What is $$\lim \frac{|S\cap [1,x]|}{x}?$$

erdos_1074.parts.iii. Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$, then does $$\lim \frac{|P\cap [1,x]|}{\pi (x)}$$ exist?

erdos_1074.parts.iv. Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$, then what is $$\lim \frac{|P\cap [1,x]|}{\pi (x)}?$$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1074» (4 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1074.parts.i : answer(sorry) ↔ ∃ c, EHSNumbers.HasDensity c

theorem erdos_1074.parts.ii : EHSNumbers.HasDensity answer(sorry)

theorem erdos_1074.parts.iii : answer(sorry) ↔ ∃ c, PillaiPrimes.HasDensity c {p | p.Prime}

theorem erdos_1074.parts.iv :
    PillaiPrimes.HasDensity answer(sorry) {p | p.Prime}

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1074. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• erdos_1074.variants.EHSNumbers_one — Regarding the first question, Hardy and Subbarao computed all EHS numbers up to 2^10, and write "...if this trend conditions we expect [the limit] to be around…

References

erdosproblems.com/1074

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.