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Claim. For every $\alpha\in(0,1)$ there is a partition of the primes $\mathbb P=P\sqcup Q$ with $\pi_P(x)\sim\alpha\,x/\log x$ and $\pi_Q(x)\sim(1-\alpha)\,x/\log x$ such that the set of integers all of whose prime divisors lie in $P$ has the translation property of Problem 675: for every $n$ some shift $t_n\ge1$ preserves membership of every $a\le n$. The claim's summary states the set for integers greater than $1$; asked in the thread whether that restriction matters, the claimant answered that it does not and updated remark 3 of the write-up to make that explicit.

Submission note. Posted to erdosproblems.com as a proof claim by Liam Price (account Leeham) on 27 July 2026, giving "GPT 5.6 Sol Pro" as the AI used:

GPT 5.6 Sol Pro proves the stronger statement that, for every $\alpha\in(0,1)$, there exists a partition of the primes

$$\mathbb{P}=P\sqcup > Q, \qquad \pi_P(x)\sim \alpha\frac{x}{\log x}, \qquad \pi_Q(x)\sim > (1-\alpha)\frac{x}{\log x},$$

such that the set of integers greater than $1$

all of whose prime divisors lie in $P$ has the translation property. In particular, taking $\alpha=\tfrac12$ answers the second question in the affirmative.

Covers. The second question of the problem, answered yes with $\alpha=1/2$: the problem asks for a partition in which each part has $\gg x/\log x$ primes up to $x$, and the claimed partition has each part of a prescribed positive proportion, which is stronger. The first question, whether the sums of two squares have the translation property, and the third, how fast the minimal shift grows for the squarefree numbers, are not addressed by this claim.

Claimant. Liam Price submitted the claim and credits GPT 5.6 Sol Pro with the proof, as the site's proof-claims tab records. The write-up is a read-only Overleaf document; this page states the claim as the site's summary and the claim's thread give it, and the argument has not been checked.

Standing. Claimed. The site's label is OPEN (proof-claims thread accessed 2026-10-06). The claim's thread holds one exchange, a question whether the restriction to integers greater than $1$ matters and the claimant's answer of 28 July 2026 that it does not, and no check, acceptance or objection; there is no refereed version.