Consecutive Integers with Restricted Prime Divisors

Definitions and mathematical statement

Let $\mathbb{P}$ denote the set of positive prime integers, and define \[ T=\{m\in{\mathbb{Z}}_{>0}:\ \forall p\in\mathbb{P},\ p\mid m\Longrightarrow p\not\equiv 1\pmod 4\}. \] Thus the permitted prime divisors are $2$ and the primes congruent to $3\pmod 4$. In particular, $1\in T$, since it has no prime divisors.

Does $T$ contain infinitely many pairs of consecutive positive integers? Precisely, is it true that \[ \forall B\in{\mathbb{Z}}_{>0}\ \exists n\in{\mathbb{Z}}_{>0}:\quad n\ge B,\qquad n\in T,\qquad n+1\in T? \] The supplied record labels this question open. That label is supplied curation, not an independently verified mathematical or literature conclusion. The bibliographic year qualification is recorded below; it does not change the stated mathematical question.

Short English statement

Are there infinitely many positive integers $n$ such that neither $n$ nor $n+1$ has a prime divisor congruent to $1\pmod 4$? The prime $2$ is allowed, and $1$ satisfies the restriction vacuously.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30002436$, title "Consecutive Integers with Restricted Prime Divisors," exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2013.
• Supplied source citation: Analytic Number Theory (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/51, pp. 2963--3037. https://doi.org/10.4171/owr/2013/51. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The supplied literature-source entry, labeled "original OWR report," gives this same link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says that the source poses the consecutive-pair assertion as a question, and reports that no primary proof or disproof was located in its search. This assessment is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent checking.
• The supplied citation labels the report 2014, while the proposed year is 2013 and the DOI contains 2013. These bibliographic details are retained as supplied without reconciliation.