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Claim. Octavio A. Agustín-Aquino and José Hernández Santiago, Infinite tetrads of congruent factorials, prove (Main Theorem) that there are infinitely many primes $p$ with

$$|\{1\le n\le p-1: n!\equiv1\pmod p\}|\ge4.$$

The construction takes an odd $q\ge3$ and a prime divisor $p$ of $q!-1$ distinct from $q+2$ and $2q+1$. Then $1!\equiv q!\equiv1$, and by Wilson's theorem $(p-1-n)!\equiv1$ whenever $n$ is odd and $n!\equiv1$, so $(p-2)!\equiv(p-1-q)!\equiv1$ as well; the four values $1$, $q$, $p-1-q$ and $p-2$ are distinct. Taking $q=6t+1$ makes $q+2$ and $2q+1$ multiples of $3$, and since every prime divisor of $q!-1$ exceeds $q$, letting $q$ grow gives infinitely many such $p$. Ordered, the four values $1<a<b<c$ bound three adjacent intervals $(1,a]$, $(a,b]$ and $(b,c]$, and the product over each is a quotient of two of the four factorials, so it is $1$ modulo $p$. The note's acknowledgment states that the valid-tetrad lemma and the main theorem were found with ChatGPT 5.5 Pro and that the write-up is the authors'. A remark added in the pinned revision cites a related construction in Hardy and Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly (2002), Remark 2.8, p. 556, which a forum reply pointed out. The note was posted on the site's forum on 3 June 2026; the pinned revision of 4 June 2026 adds that remark and a corrected bibliography.

Covers. The case $k=3$ of Problem 1056, for infinitely many primes $p$, under the sources' reading recorded in the problem page's Formulation; since the common residue is $1$, the interval $\{1\}$ gives a fourth interval under the site's wording. The case $k=3$ itself was already settled by [[problems/diophantine_problems/E1056/claims/1983_01_01_makowski|Mąkowski's example]].

Formalization. The repository's tetrads.lean declares itself a Lean formalization of the note's main theorem and proves infinitely_many_tetrad_primes_unbounded, that for every $N$ some prime $p>N$ has a tetrad. It is third-party Lean that this corpus has not built or audited, so no formalized evidence is listed.

Standing. Claimed. The note is posted in a public repository and on the site's forum, with no review or refereed publication recorded, and the site labels the problem OPEN. Nothing here is independently reviewed by this project.