Problem 854

../

claims/: The 3 claim pages of Problem 854, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.

If $1=a_1<a_2<\cdots a_{\phi(n_k)}=n_k-1$ is the sequence of integers coprime to $n_k$, then estimate the smallest even integer not of the form $a_{i+1}-a_i$. Are there

$$\gg \max_i (a_{i+1}-a_i)$$

many even integers of the form $a_{j+1}-a_j$?

Status. Open. The site's label is OPEN (page last edited 4 November 2025). Its proof-claim tab carries one partial claim, submitted 2026-08-20 by the forum account DottedCalculator, the claimant, with a five-page write-up signed by the AI system GPT 5.6 Sol: for every large $k$, every even number up to a constant times $p_k$ is a difference $a_{i+1}-a_i$, so the smallest even integer that is not such a difference is $\gg p_k\sim k\log k$ and $\gg p_k$ distinct even differences occur; the write-up says that it does not settle the displayed comparison with $\max_i(a_{i+1}-a_i)$. The claim is recorded on its claim page without adoption: no comment, review or outside record was found, and nothing is reviewed here. The write-up names two earlier public lower bounds, each recorded as a claimed partial result: Ziller's 2020 preprint shows that $2,4,\ldots,2k$ all occur as differences, and White's working report of 27 July 2026, written with Claude, proves that the smallest even non-difference is at least $(e^{\gamma}-o(1))k\log\log k$. A partial claim leaves the standing open.

Source. erdosproblems.com/854, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #854, https://www.erdosproblems.com/854.

References.

• [Ob1] P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various).

Formalization. None recorded.

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.

• erdos_1985_my_problems_number_theory_i_would
• erdos_1985_my_problems_number_theory_i_would / question_p80_totative_gaps