Problem 235

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claims/: The 1 claim page of Problem 235, one per claimant's result; the problem's standing derives from them.

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Statement. Let $N_k=2\cdot 3\cdots p_k$ and ${a_1<a_2<\cdots <a_{\phi(N_k)}}$ be the integers $<N_k$ which are relatively prime to $N_k$. Then, for any $c\geq 0$, the limit

$$\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}$$

exists and is a continuous function of $c$.

Status. Proved. Hooley's Theorem 1 ([Ho65], refereed) shows that, as $n/\phi(n)\to\infty$, the proportion of gaps below $cn/\phi(n)$ between the integers prime to $n$ tends to $1-e^{-c}$, uniformly for $c$ in any fixed range bounded away from $0$ and $\infty$; the problem's limit, which counts gaps up to $cN_k/\phi(N_k)$, follows for every $c>0$ and is $0$ at $c=0$, so it is the continuous function $1-e^{-c}$; the site records the problem as solved by Hooley, and the claim page carries the acceptance, together with a 2026 Lean formalization of Hooley's theorem in a public repository, registered by no outside record and neither built nor audited here.

Source. erdosproblems.com/235, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #235, https://www.erdosproblems.com/235.

References.

• [Ho65] Hooley, Christopher, On the difference between consecutive numbers prime to $n$. II. Publ. Math. Debrecen 12 (1965), 39--49, doi:10.5486/pmd.1965.12.1-4.06.

Formalization. No statement in formal-conjectures (the site shows no formalized statement, and the community database records the problem unformalized). The file src/latest/ErdosProblems/Erdos235.lean of plby/lean-proofs states erdos_235, the existence of a continuous limit of the problem's quotient for every $c\ge0$, with the limit $1-e^{-c}$; Hooley's claim page above links it at its pinned commit and records what it rests on.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• hooley_1965_difference_between_consecutive_numbers_prime
• hooley_1965_difference_between_consecutive_numbers_prime / theorem_1
• hooley_1965_difference_between_consecutive_numbers_prime / theorem_2