---

Statement

This is a heuristic model, not a proved result. Socialist primes are defined on the page for (2.6).

Model (p. 4). Treat the residues of $2!,\ldots,(p-1)!$ modulo $p$ as random nonzero residues, and use that $(k+1)!\not\equiv k!$ for $2\le k\le p-2$. The estimated probability that $p$ is socialist is then

$$W_p=\frac{(p-2)!}{(p-2)^{p-3}}.$$

(4.1) (p. 4). By Stirling's approximation, $W_p\le(p-2)^{3/2}e^{3-p}$. The paper calls this "just a rough upper bound" (p. 4, quoted): conditions (1.1) and (1.2) would lower it by some factor; treating $!^kp\bmod p$, $k=1,\ldots,p-2$, as independent and random, (2.7) would suggest $W_p\approx p^{2-p}$, but the congruence $!^{2k}p\equiv\,!^{p-2k-1}p\pmod p$ for $1\le k\le(p-3)/2$ and odd primes $p$ (p. 5) means $W_p$ should be larger than that.

Expected count (p. 5). Estimating the number of socialist primes in $[a,b]$ by $\sum_{a\le p\le b}W_p$, approximated by an integral, the paper derives

$$\sum_{a\le p\le b}W_p<e^3a^{3/2-a}\sqrt{\ln a},$$

and so expects no more than $e^3a^{3/2-a}\sqrt{\ln a}$ socialist primes greater than $a$. Combined with the search below $10^{11}$, it estimates the probability that socialist primes exist at less than $10^{-10^{12}}$.

Proof pointer

Pp. 4--5: Stirling's bound $k!\le e\,k^{k+1/2}e^{-k}$ gives (4.1); the count replaces the sum over primes by $\int_a^b W_{t\ln t}\,dt$ and bounds the integrand by the derivative of $-t^{3/2-t}(\ln t)^{1/2}$. The replacement of the sum by an integral is heuristic.

Read depth

Claims checked: the model, (4.1) and the expected-count estimate were read on the arXiv v1 print, pp. 4--5. Nothing here is independently reviewed.

Dependencies

Condition (2.7) and the [[factorials_binomials/andrejic_2016_distinct_residues_factorials/computation_p6|search below $10^{11}$]], for the remarks only.

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: a heuristic estimate under which the extreme case $\lvert A_p\rvert=p-2$ with $p>5$ (socialist primes; see the page for (2.6)) is expected to be rare; it proves nothing about $A_p$.