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Claim. Jens Kruse Andersen posted on Prime Puzzles problem 27 on 4 May 2007 the least primes admitting $k$ adjacent intervals of consecutive integers whose products are each $1$ modulo $p$, for $k=10,\ldots,14$: $27901$, $52163$, $778699$, $2374649$ and $10428007$, each with its intervals. For $k=14$ the intervals are $[816489,1251081]$, $[1251082,3384225]$, $[3384226,4112650]$, $[4112651,4237275]$, $[4237276,4431559]$, $[4431560,4467010]$, $[4467011,4835062]$, $[4835063,7328694]$, $[7328695,7385077]$, $[7385078,7415726]$, $[7415727,8460938]$, $[8460939,8689396]$, $[8689397,9295594]$ and $[9295595,9661614]$, modulo the prime $10428007$. The method is a search over the residues $q!\bmod p$ for $q<p$: if $q_1!\equiv q_2!\not\equiv0$, the product over $(q_1,q_2]$ is $1$ modulo $p$, and a residue occurring $k+1$ times gives $k$ adjacent intervals. A direct computation confirms that $10428007$ is prime and that all fourteen products are $1$ modulo it. The first $k$ of these intervals form a witness for each $k$ from $2$ to $14$.

Andersen submitted the five primes to OEIS A060427, whose terms run through $a(14)=10428007$; the entry, created by Jason Earls in 2001, lists only the least primes. The smaller cases on the same Prime Puzzles page are C. Rivera's recomputation of the least primes for $k\le9$ ($11$, $17$, $23$, $71$, $599$ and $3011$), following Landon Curt Noll's submission of the problem to the site in 1999. A forum post by Kenta Kitamura of 21 June 2026, prepared with the assistance of Codex 5.5 and ChatGPT 5.5 Pro, restates the $k=14$ witness as fifteen factorials congruent to $8978998$ modulo $10428007$ and the fourteen intervals they bound; it is a thread post restating this result and has no page of its own.

Covers. Every case $2\le k\le14$ of Problem 1056: such a prime exists for each of these $k$, under the sources' reading recorded in the problem page's Formulation.

Standing. Claimed. The witnesses are finite computations, but they were posted on a puzzle site and in the OEIS, with no review or refereed publication recorded, and the site labels the problem OPEN. Nothing here is independently reviewed by this project.