Problem 972

Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha\rfloor$ is also prime?

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. No proof, disproof, preprint or proof claim for the exact statement was found in the search whose scope the Current assessment records. The one-prime statement is classical and, for $\alpha$ of finite type, quantitative (Banks and Shparlinski's Theorem 5.4); the two-prime statement is known for almost all $\alpha$ in the sense of Lebesgue measure (Li and Pan, 2009, second-hand from the arXiv abstract), which leaves every individual $\alpha$ , hence the question, open; the thread's judgment puts it at the difficulty of the twin prime conjecture. This is a bounded negative finding, not a certificate of openness.

Er65b: Erdős, P., Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196--244. Printed p. 210; the site's key gives no page. Erdős's reference [24] there groups Turán's 1937 Acta Szeged paper on primes in arithmetic progressions, Erdős's 1949 paper on applications of Brun's method (Acta Szeged 13, 57--63) and Vinogradov's 1948 paper [Vi48] (reference list, printed pp. 240--241). Library home: Erdős 1965.
Vi48: Vinogradov, I. M., On an estimate of trigonometric sums with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 225--248 (Russian; received 15 January 1948). Theorems 1--4 (hypotheses and conclusions), printed pp. 234--235, 243 and 246--248. Library home: Vinogradov 1948.
BaSh07: Banks, W. D. and Shparlinski, I. E., Prime numbers with Beatty sequences. arXiv:0708.1015v1 (7 August 2007); Colloq. Math. 115 (2009), no. 2, 147--157, doi:10.4064/cm115-2-1 (Crossref record accessed; the journal text not held or compared; locators are the preprint's). Theorem 5.1 (p. 7), Theorem 5.4 and Corollaries 5.5--5.6 (p. 11). Library home: Banks and Shparlinski 2007; result page Banks and Shparlinski 2007, Theorem 5.4 (p. 11).
LiPa09: Li, H. and Pan, H., Primes of the form $\lfloor\alpha p+\beta\rfloor$ . J. Number Theory 129 (2009), no. 10, 2328--2334, doi:10.1016/j.jnt.2009.03.009 (Crossref record accessed); arXiv:0803.1740v3 (5 April 2008), known here by its abstract only. Not held; a lead by identifier.
So21: Song, Y., A note on primes of the form $\lfloor\alpha p+\beta\rfloor$ . J. Number Theory 225 (2021), 1--17, doi:10.1016/j.jnt.2021.01.005 (Crossref record accessed; known here by its title only). A lead by identifier.
Di20: Dimitrov, S. I., On the distribution of $\alpha p$ modulo one over Piatetski-Shapiro primes. arXiv:2005.05008 (v2, 1 May 2025; abstract only): for irrational $\alpha$ , real $\beta$ and $1<c<12/11$ , infinitely many primes $p=[n^c]$ with $\|\alpha p+\beta\|\ll p^{(11c-12)/(26c)}\log^6p$ ; a one-prime distribution result, context only.

Problem 972

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Proof claims

Discussion