Hardy 2002 modified problem pillai related questions

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problem_a: Hardy and Subbarao's open Problem A, raised in discussion with Erdős, asks whether the number of Pillai primes up to x divided by the number of primes up to x has a limit, with computations suggesting a value near 0.5 to 0.6 if it exists.

problem_b: Hardy and Subbarao's open Problem B, raised in discussion with Erdős, asks whether the count of EHS numbers up to x divided by x has a limit and what it is; the authors believe the density exists and equals one.

problem_f: Erdős's Problem F*, as printed by Hardy and Subbarao, asks whether the number A(x) of composite numbers u below x with n!+1 divisible by u satisfies A(x) = o(x^epsilon), with 25, 121 and 721 given as examples.

problem_g: Hardy and Subbarao's open Problem G, raised in discussion with Erdős, on the least f(p) with f(p)!+1 divisible by p: they believe f(p) = p-1 for infinitely many but o(x/log x) primes p up to x, and Erdős believed f(p)/p tends to 0 for almost all p.

theorem_2_1: Hardy and Subbarao's theorem that infinitely many primes p admit an integer n with n!+1 divisible by p and p not congruent to 1 mod n, the primes Definition 2.9 calls Pillai primes.

theorem_2_12: Hardy and Subbarao's theorem that infinitely many natural numbers m admit a prime p dividing m!+1 with p not congruent to 1 mod m, the numbers Definition 2.11 calls EHS numbers.

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G. E. Hardy, M. V. Subbarao, A Modified Problem of Pillai and Some Related Questions. The American Mathematical Monthly 109 (2002), 554-559. doi:10.2307/2695445. The JSTOR PDF prints "© THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 109" at the foot of printed p. 554 (PDF p. 2, read on the page image), and its JSTOR cover sheet's "you may use content in the JSTOR archive only for your personal, non-commercial use" is the platform's notice, every other right reserved.

Theorem 2.1 (printed p. 555) shows there are infinitely many primes p (called Pillai primes) for which some n has n!+1 = 0 mod p while p is not 1 mod n, answering Problem 1.2 (p. 554), a modified form of Pillai's question, and Theorem 2.12 answers Problem 1.4 in the same way; the paper notes that Erdos and, independently, Subbarao had found solutions in 1993 and gives another proof. That proof takes the largest prime p dividing (10K+7)!+1; if p = 1 mod (10K+7), it uses Wilson's theorem to get (p-10K-8)!+1 = 0 mod p, and rules out p = 1 mod (p-10K-8) by an elementary A+B = AB argument. Theorem 2.12 (p. 556) deduces that the companion set S of such integers m (EHS numbers) is also infinite, and Remark 2.13 notes the converse implication. Section 3 (p. 557) lists open problems A through H, including the density of Pillai primes among primes, the density of EHS numbers, and the growth of the least n with n!+1 = 0 mod p. Problem F*, starred as original to Erdos, defines A(x) as the number of composite u < x with n!+1 = 0 mod u (examples 25, 121, 721) and asks: "Is $A(x)=o(x^\epsilon)$?" (p. 557). Read for every $\epsilon>0$, this is the question $A(x)\le x^{o(1)}$ of problem 1073, and it is the source statement behind that problem: the paper poses it and offers no bound or partial result toward it.

Source: https://www.math.ualberta.ca/~subbarao/documents/2002_Pillai.pdf.

Bears on. #1072: Problem G (p. 557) states the problem's two questions as beliefs, the authors' that $f(p)=p-1$ for infinitely many primes $p$ and Erdős's that $f(p)/p\to0$ for almost all $p$; the paper proves nothing about them. #1073: Problem F* (p. 557), read for every $\epsilon>0$, is the problem's question; the paper poses it and proves nothing toward it. #1074: the problem's sets $S$ and $P$ are the paper's EHS numbers and Pillai primes, which Theorem 2.12 and Theorem 2.1 show are infinite; the problem's density questions are posed in Problem B and, for the existence of the limit only, in Problem A, with numerical data and no proof.

Results. Claims checked on the page images of the print; nothing here is independently reviewed.

• Theorem 2.1 (p. 555), with Definition 2.9 (p. 556): there are infinitely many Pillai primes, primes $p$ with $n!+1\equiv0\pmod p$ and $p\not\equiv1\pmod n$ for some $n$.
• Theorem 2.12 (p. 556), with Definition 2.11: the set $\mathcal S$ of EHS numbers, the $m$ admitting such a prime, is infinite.
• Problem A (p. 557): does $\pi(\mathcal P,x)/\pi(x)$ have a limit?
• Problem B (p. 557): does the density of the EHS numbers exist, and what is it?
• Problem F* (p. 557): is $A(x)=o(x^\epsilon)$ for the count $A(x)$ of composite $u<x$ with $n!+1\equiv0\pmod u$?
• Problem G (p. 557): the least $f(p)$ with $f(p)!+1\equiv0\pmod p$ is believed to equal $p-1$ for infinitely many primes $p$, but probably for only $o(x/\log x)$ primes $p\le x$, and Erdős believed $f(p)/p\to0$ for almost all $p$.

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