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Claim. Andrzej Mąkowski, On a number-theoretical problem of Erdős, Elem. Math. 38 (1983), Heft 4, 101--102, a short communication in the journal's Kleine Mitteilungen section, answers the case $k=3$ with the prime $17$:

$$2\cdot3\cdot4\cdot5\equiv6\cdot7\cdot8\cdot9\cdot10\cdot11\equiv12\cdot13\cdot14\cdot15\equiv1\pmod{17}.$$

The three intervals $[2,5]$, $[6,11]$ and $[12,15]$ are adjacent, and each product is $1$ modulo $17$, as a direct computation confirms. The site's commentary credits this example to the note. Guy's A15 (3rd ed., 2004, p. 54), which cites the note, reports that Mąkowski sent the examples of rows $n=5$ and $n=6$ of the Noll--Simmons table and remarked that tables of indices can be used to find others. Row $6$ is the prime $23$, with $2\cdot3\cdot4\equiv5\cdots8\equiv9\cdot10\cdot11\equiv12\cdots21\equiv1\pmod{23}$, four adjacent intervals and so the case $k=4$; Prime Puzzles problem 27 also credits Mąkowski with $k=3$ and $k=4$ on Guy's authority. Guy does not say that the $p=23$ example is printed in the note, so this page covers only $k=3$; the case $k=4$ is covered by [[problems/diophantine_problems/E1056/claims/2007_05_04_andersen|Andersen's claim page]] in any case. The intervals are stated under the sources' reading recorded in the problem page's Formulation; since the common residue is $1$, the interval $\{1\}$ added in front gives four intervals under the site's wording.

Covers. The case $k=3$ of Problem 1056: such a prime exists, namely $p=17$.

Formalization. The formal-conjectures file for the problem proves this example as erdos_1056.variants.k3 by decide (1056.lean). That is related Lean this corpus has not built, so no formalized evidence is listed.

Acceptance. Refereed: the note appeared in Elemente der Mathematik 38 (1983), pp. 101--102, as the zbMATH record (Zbl 0521.10003) gives it. The site labels the problem OPEN, so its credit is not counted as review. Nothing here is independently reviewed by this project.