Euclid-Type Expressions for Primes Congruent to Three Modulo Four

Definitions and mathematical statement

Let $p_1<p_2<\cdots$ enumerate all positive prime numbers, and let $q_1<q_2<\cdots$ enumerate all positive primes congruent to $3\pmod 4$. For each integer $k\geq 1$, define \[ P_k=\prod_{i=1}^{k}p_i,\qquad Q_k=\prod_{i=1}^{k}q_i. \] For an integer $n\geq 2$, define its least prime divisor by \[ \operatorname{lpf}(n)=\min\{\ell:\ell\text{ is prime and }\ell\mid n\}. \] The supplied question asks whether one can construct a Euclid-type expression in $Q_k$ whose value $E_k$ satisfies \[ E_k\in{\mathbb{Z}}_{\geq 2},\qquad \operatorname{lpf}(E_k)=q_{k+1} \quad\text{for every integer }k\geq 1. \] Here $E_k$ denotes the value of the requested expression; no admissible class of expressions is specified by the source record. In particular, the meanings of "Euclid-type" and "expression in $Q_k$", the permitted auxiliary inputs, and the required uniformity in $k$ are unresolved. Consequently, the displayed condition records the requested least-prime-divisor property but does not constitute a fully specified existence problem until those restrictions are supplied. The product $P_k$ is included in the source setup, but no role for it in the requested construction is stated.

Short English statement

The question asks for a Euclid-type construction from the product of the first $k$ primes congruent to $3\pmod 4$, with least prime divisor exactly the next such prime. The permitted constructions are unspecified, so the intended existence question requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30003280$, source problem code catalogue entry: supplied complete source record for "Euclid-Type Expressions for Primes Congruent to Three Modulo Four", in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Analytic Number Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/53, pp. 2975--3030. https://doi.org/10.4171/owr/2016/53. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, "OWR 53 report": https://publications.mfo.de/bitstream/handle/mfo/3557/OWR_2016_53.pdf?isAllowed=y&sequence=1.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and reports that no authoritative later construction or impossibility proof was found. This is supplied dated curation, not an independently verified literature assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with the qualification that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This registry label does not establish independent verification of the formulation or mathematical review of this draft.