---

Statement

Setting (p. 1). A prime $p>5$ is a socialist prime (Trudgian's term, which the paper adopts) when the residues of $2!,3!,\ldots,(p-1)!$ modulo $p$ are all distinct. Kurepa's left factorial is $!n=0!+1!+\cdots+(n-1)!$.

Let $p$ be a socialist prime, and let $r$ be the one nonzero residue modulo $p$ that is not among $2!,\ldots,(p-1)!$ (there are $p-2$ distinct values among $p-1$ nonzero residues). The paper derives in Section 2 (pp. 2--3):

• (2.4) (p. 2). $p\equiv1\pmod 4$ and $\bigl(\bigl(\tfrac{p-1}{2}\bigr)!\bigr)^2\equiv-1\pmod p$.
• (2.5) (p. 2). $r\equiv-\bigl(\tfrac{p-1}{2}\bigr)!\pmod p$, and consequently $(p^2-1)/8$ is odd, so $p\equiv5\pmod 8$.
• (2.6) (p. 2). $\bigl(\tfrac{p-1}{2}\bigr)!\equiv\;!p-2\pmod p$, and hence the necessary condition

$$(!p-2)^2\equiv-1\pmod p.$$

The value of the missing residue and $p\equiv5\pmod 8$ were already proved by Rokowska and Schinzel (1960), as the paper recalls on p. 1 with (1.1); the paper rederives them here. Condition (2.6), linking socialist primes to Kurepa's left factorial, is the paper's new condition.

Proof pointer

P. 2. Wilson's theorem gives $(p-1)!\equiv-1$ and $(p-2)!\equiv1$, and the reflection $(p-k)!\,(k-1)!\equiv(-1)^k\pmod p$ for $1\le k\le p$ gives $\bigl(\bigl(\tfrac{p-1}{2}\bigr)!\bigr)^2\equiv(-1)^{(p+1)/2}$. Distinctness forbids $\bigl(\tfrac{p-1}{2}\bigr)!\equiv\pm1$, which forces $p\equiv1\pmod 4$ and (2.4). Writing $(p-1)!$ as $r$ times the product of all $k!$, $2\le k\le p-1$, and pairing factorials by the reflection, gives $r\equiv(-1)^{(p^2-1)/8}\bigl(\tfrac{p-1}{2}\bigr)!$; since $r$ differs from $\bigl(\tfrac{p-1}{2}\bigr)!$, (2.5) follows. Finally the residues $2!,\ldots,(p-1)!$ together with $r$ run over $1,\ldots,p-1$, whose sum is $0$ mod $p$; this gives $\bigl(\tfrac{p-1}{2}\bigr)!\equiv\;!p-2$, and (2.4) turns it into (2.6).

Read depth

Claims checked: the definitions and (2.1)--(2.6) were read clause by clause on the arXiv v1 print, pp. 1--2, and the derivation on p. 2 was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: Wilson's theorem.

Source. V. Andrejić and M. Tatarevic, On distinct residues of factorials, arXiv:1603.04086v1 (2016); published in Publ. Inst. Math. (Beograd) (N.S.) 100(114) (2016), 101--106. Labels and pages here are those of the arXiv v1 print; the edition read is named on the source card.

Bears on

• Problem 478: for $p\ge5$ one has $1!\equiv(p-2)!\pmod p$, so the problem's $A_p$ has at most $p-2$ elements, with equality exactly when the residues of $2!,\ldots,(p-1)!$ are distinct: at $p=5$, and for $p>5$ exactly when $p$ is a socialist prime (an observation of this page, not of the paper). The conditions here constrain only that extreme case; they say nothing about the size of $A_p$ in general or about the asymptotic the problem asks for.