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Claim. Theorem 9.1 of T. N. Shorey and R. Tijdeman, Arithmetic properties of blocks of consecutive integers, in From Arithmetic to Zeta-Functions (Springer, 2016), 455–471, arXiv:1612.05438 (Shorey and Tijdeman (2016)). It assumes Baker's explicit abc conjecture, their Conjecture 9.1: for pairwise coprime positive integers $a,b,c$ with $a+b=c$, $c<\frac65R\,(\log R)^{w}/w!$, where $R$ is the product of the distinct primes dividing $abc$ and $w$ is their number. Under that hypothesis the theorem reads: "there are no positive integers $n_1<n_2$ such that for $i=0,1,2$ the numbers $n_1+i$ and $n_2+i$ have the same prime divisors" (Theorem 9.1). So, under the hypothesis, the answer to Problem 850, read for positive integers as the problem page's Formulation records, is no.

The proof applies the consequence $c<R^{7/4}$ of the hypothesis, due to Laishram and Shorey, to the identity $(n_2+1)^2-n_2(n_2+2)=1$. Every prime dividing $n_2(n_2+1)(n_2+2)$ divides $n_2-n_1$, so $R\le n_2-n_1<n_2$, which gives $n_2^2<n_2^{7/4}$, a contradiction.

Hypothesis. Baker's explicit abc conjecture is unproved, so the claim gives no unconditional answer.

Standing. The sources are a chapter in an edited volume and an arXiv preprint, which describes itself as a corrected and extended version of the chapter; no journal publication is recorded, so refereed is not listed. The site's commentary credits the conditional answer to Shorey and Tijdeman, but on a problem the site labels OPEN that commentary is not acceptance, so the claim stays claimed. The page is dated by the arXiv posting of 16 December 2016, before the chapter's online date of 31 December 2016.

Depends on. Nothing on this wiki; the hypothesis is stated above.