---

Statement

Problem A (p. 557). Let $\pi(x)$ count the primes and $\pi(\mathcal P,x)$ the Pillai primes (the primes of Definition 2.9) up to $x$. The paper asks whether $\pi(\mathcal P,x)/\pi(x)$ has a limit as $x\to\infty$.

The authors add that their table (Section 4, item (ii), p. 558: the ratio at ten selected Pillai primes, from $0.111111$ at $23$ to $0.530053$ at $44987$) suggests that the limit, if it exists, is perhaps between $0.5$ and $0.6$, while they see no reason the ratio should not tend to $1$, very slowly and not monotonically.

Section 3 (p. 557) says the problems not marked with an asterisk, except Problem H, were raised in discussions with Erdős; Problem A is not starred.

Proof pointer

An open problem; the paper proves nothing about it beyond the computed table.

Read depth

Claims checked: Problem A and the table of Section 4 were read clause by clause on the page images of the print. The table was not recomputed. Nothing here is independently reviewed.

Dependencies

• Theorem 2.1: the Pillai primes are infinite in number.

Source. G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109 (2002), no. 6, 554--559, doi:10.2307/2695445; the edition read is named on the source card.

Bears on

• Problem 1074: the problem's second question asks whether $\lvert P\cap[1,x]\rvert/\pi(x)$ has a limit and what it is. With the site's $P$ equal to the paper's $\mathcal P$, Problem A asks the first part; it does not ask for the value but guesses it from the table. The paper gives numerical data only.