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Claim. In a letter dated 31 October 1979, as Guy reports in problem A15 of Unsolved Problems in Number Theory, Erdős observed that

$$3\cdot4\equiv5\cdot6\cdot7\equiv1\pmod{11},$$

so the adjacent intervals $[3,4]$ and $[5,7]$ have products $12$ and $210$, both $1$ modulo $11$. In the same letter he asked for the least prime with three such products and suggested that primes exist for any number of congruent products. The problem first appeared in the first edition of Guy's book (1981) as A15, as Landon Curt Noll states on Prime Puzzles problem 27, and the page is dated by that edition; the third edition (2004, p. 54) carries it under the same number, see [[../library/number_theory/guy_2004_unsolved_problems_number_theory/_index|Guy's card]]. The site's commentary credits the case $k=2$ to this observation. The formal-conjectures file for the problem proves the example as erdos_1056.variants.k2 by decide (1056.lean), Lean this corpus has not built.

Covers. The case $k=2$ of Problem 1056: such a prime exists, namely $p=11$, the least one under the sources' reading recorded in the problem page's Formulation.

Standing. Claimed. The example is a finite computation that anyone can check, but it was published in a letter as Guy reports it, and the site labels the problem OPEN; no review or refereed publication of the example itself is recorded. Nothing here is independently reviewed by this project.