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Statement

Socialist primes and Kurepa's left factorial $!p$ are defined on the page for (2.6); $r_p$ denotes $!p\bmod p$.

Below $2^{34}$ (pp. 2--3). Using the residues $r_p$ for all primes $p<2^{34}$, recorded in the authors' earlier work (V. Andrejić and M. Tatarevic, Searching for a counterexample to Kurepa's conjecture, arXiv:1409.0800, then to appear in Math. Comp.), the paper reports that the only primes $p<2^{34}$ with $p\mid(r_p-2)^2+1$ are $5$, $13$, $157$, $317$, $5449$ and $5749$, with $r_5=4$, $r_{13}=10$, $r_{157}=131$, $r_{317}=205$, $r_{5449}=4816$ and $r_{5749}=808$. By (2.6) there are therefore no socialist primes with $5<p<2^{34}$. Section 5 (p. 5) restates this as checking (2.6) for $10^9<p<2^{34}$, the range below $10^9$ having been covered by Trudgian. The paper also remarks (p. 3) that no small $p\equiv3\pmod 4$ divides $(r_p-2)^2+1$.

Below $10^{11}$ (abstract, p. 1; Section 5, pp. 5--6). The paper reports that there are no socialist primes less than $10^{11}$. The search beyond $2^{34}$ examined only primes satisfying the Rokowska--Schinzel conditions (1.1) and, for each, looked for a pair $2\le i<j\le p-1$ with $i!\equiv j!\pmod p$ by a birthday-collision search, of expected cost about $\sqrt{\pi p/2}$ factorial evaluations per prime; the run over all $p<10^{11}$ took slightly over one day on one CPU (p. 6).

Both results are machine computations reported by the authors; the paper gives the method, not a certificate, and they have not been rerun here.

Read depth

Claims checked: the reported values and ranges were read on the arXiv v1 print, pp. 1--3 and 5--6. The computations are not checked. Nothing here is independently reviewed.

Dependencies

Condition (2.6) for the range below $2^{34}$. External inputs: the authors' table of $!p\bmod p$ for $p<2^{34}$ (arXiv:1409.0800), the Rokowska--Schinzel conditions (1.1), and Trudgian's search below $10^9$.

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 $p>5$ the problem's $A_p$ has at most $p-2$ elements, since $1!\equiv(p-2)!\pmod p$, with equality exactly when $p$ is a socialist prime (an observation of this page, not of the paper). The search therefore reports $\lvert A_p\rvert\le p-3$ for every prime $5<p<10^{11}$; it says nothing about the asymptotic size of $A_p$.