Problem 972

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

Formulation. The site's wording (the page carries no last-edited date). The question asks, for every fixed irrational $\alpha>1$, for infinitely many primes $p$ such that the Beatty-sequence value $\lfloor p\alpha\rfloor$ is prime. Erdős's 1965 print (p. 210) poses it as "whether there are infinitely many primes $p$ for which $[p\alpha]=q$" right after recording that for every irrational $\alpha>1$ the equation $[n\alpha]=p$ has infinitely many prime solutions, the one-prime statement. The thread notes that the question makes sense for every positive $\alpha$ outside $\mathbb N$ and that for $\alpha=1/2$ it is the infinitude of Sophie Germain primes; those are variants of the site's $\alpha>1$. The formal-conjectures file encodes the site's statement exactly. The heuristic count of such primes up to $x$ is of order $x/(\log x)^2$, as for twin primes (the thread), a heuristic and not a result.

Status. 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.

Source. erdosproblems.com/972, accessed 2026-09-18: the problem page (OPEN, with the site's note that no finite computation can settle it; no last-edited date; source key [Er65b]; commentary citing [Vi48]), its two-comment discussion thread (8 October 2025) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #972, https://www.erdosproblems.com/972, accessed 2026-09-18.

References.

• [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: erdos_1965_recent_advances_current_problems_number_theory.
• [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_estimate_trigonometric_sums_prime_numbers.
• [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_2007_prime_numbers_beatty_sequences; result page theorem_5_4.
• [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.

Formalization. Statement only. The file ErdosProblems/972.lean of formal-conjectures at the linked commit (main) defines primeSet (α : ℝ) : Set ℕ := {p : ℕ | Nat.Prime p ∧ Nat.Prime ⌊ (α * p) ⌋₊} and declares erdos_972 : answer(sorry) ↔ ∀ α > 1, Irrational α → (primeSet α).Infinite under category research open, with proof sorry. The community database (teorth/erdosproblems) records the problem open (31 August 2025), the statement formalized since 7 December 2025, and no formal proof. Nothing was built.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN; no last-edited date. The commentary derives the one-prime statement from Vinogradov's theorem [Vi48] that $\{p\alpha\}$ is uniformly distributed for every irrational $\alpha$: a prime $p$ has the form $\lfloor n\alpha\rfloor$ exactly when $p/\alpha\le n<(p+1)/\alpha$ for some integer $n$, that is, exactly when $\{p\alpha^{-1}\}>1-\alpha^{-1}$, and the uniform distribution of $\{p\alpha^{-1}\}$ gives infinitely many such $p$ for every irrational $\alpha>1$. The thread (8 October 2025): Tao judges the problem to be of about the difficulty of the twin prime conjecture; a second comment expects the statement for every $\alpha\in\mathbb R^+\setminus\mathbb N$, notes that for $\alpha=1/2$ it is equivalent to the infinitude of Sophie Germain primes, with an expected count $2C(1+o(1))x/\ln^2x$ for $C$ the twin prime constant, and expects order $x/\ln^2x$ for every valid $\alpha$. The proof-claim tab is empty. The community database record says open.

Origin. [Er65b], printed p. 210, recalls that Turán, using the results cited as [24], proved the uniform distribution of $p\alpha\pmod1$ for every irrational $\alpha>1$, that Vinogradov [24] later proved it without any hypothesis, and that his trigonometric-sum estimates also give a good bound for the discrepancy of the sequence; then: "It follows easily from the uniformity of distribution that for every irrational $\alpha>1$, $[n\alpha]=p$ has infinitely many solutions. As far as I know it is not known whether there are infinitely many primes $p$ for which $[p\alpha]=q$." The passage records the one-prime statement as known and poses the two-prime question; it proves nothing.

The one-prime statement (settled; not the problem). The site's argument is checked here: for $\alpha>1$ irrational and a prime $p$, the interval $[p/\alpha,(p+1)/\alpha)$ has length $\alpha^{-1}<1$ and contains an integer $n$ exactly when the fractional part of its left endpoint exceeds $1-\alpha^{-1}$, so $p=\lfloor n\alpha\rfloor$ for some $n$ if and only if $\{p\alpha^{-1}\}>1-\alpha^{-1}$, and the uniform distribution of $\{p\alpha^{-1}\}$ over the primes gives a proportion $\alpha^{-1}$ of the primes of this form. The uniform distribution of $\{p\theta\}$ for irrational $\theta$ is Vinogradov's theorem, which Erdős and the site both cite to [Vi48]. The 1948 paper does not state it: its standing notation fixes an integer constant $n\ge10$ bounding the degree of the phase (pp. 225--226); Theorem 1 (pp. 234--235) bounds $S=\sum_{p\le P}e^{2\pi ilf(p)}$ for $f(p)=a_np^n+\cdots+a_1p$ when a coefficient $a_s$ with $s\in\{2,\ldots,n\}$ has a rational approximation $a/q+\theta/(q\tau)$, $(a,q)=1$, $P^\varkappa\ll q\le\tau=P^{0.5s}$, giving $S\ll P^{1-\rho}$ for $l\le P^{2\rho_0}$, $\rho_0$ being the slightly larger exponent of Lemma 4 (p. 228), with $\rho$ explicit in $\nu=1/n$, $n$ and $\varkappa$; Theorem 3 (pp. 246--247) turns this into the count $T=\gamma\pi(P)+O(P^{1-\rho'})$ of primes $p\le P$ with $0\le\{f(p)\}<\gamma$; Theorems 2 and 4 (pp. 243, 247--248) treat smooth phases $f$ with sign and size conditions on $f^{(n-1)}$, $f^{(n)}$ and $f^{(n+1)}$ on an interval $(P_1,P_2]$. The linear phase $\alpha p$ satisfies none of these hypotheses. The paper's own references are Vinogradov's Doklady notes of 1946 and 1947 and his 1947 book on the method of trigonometric sums, where the linear case belongs; so the citation identifies the author's method and not the theorem used, a source-identity remark recorded here without consequence for the status (the one-prime conclusion is classical and not in question). For $\alpha$ of finite type the one-prime statement is also quantitative: Theorem 5.4 of [BaSh07] (p. 11) gives, for fixed real $\alpha,\beta$ with $\alpha$ positive, irrational and of finite type, a $\kappa>0$ with

$$\sum_{\substack{n\le N\\ \lfloor\alpha n+\beta\rfloor\equiv a\ (\mathrm{mod}\ q)}}\Lambda(\lfloor\alpha n+\beta\rfloor)=\alpha^{-1}\sum_{\substack{m\le\lfloor\alpha N+\beta\rfloor\\ m\equiv a\ (\mathrm{mod}\ q)}}\Lambda(m)+O\bigl(N^{1-\kappa}\bigr)$$

uniformly for $0\le a<q\le N^\kappa$, $\gcd(a,q)=1$, and its Corollary 5.6 gives, for $(a,q)=(0,1)$, $\sum_{n\le N}\Lambda(\lfloor\alpha n+\beta\rfloor)=N+O(N\exp(-c(\log N)^{3/5}(\log\log N)^{-1/5}))$, so the Beatty sequence carries the expected number of primes; Theorem 5.1 (p. 7) is the analog for primes $q\lfloor\alpha n+\beta\rfloor+a$. Read depth: claims checked; the proofs (Sections 3--5) were not checked. Finite type holds for almost all $\alpha$, not for every irrational $\alpha$, so this quantitative form covers fewer $\alpha$ than Vinogradov's theorem does qualitatively. Neither result constrains $p$ itself to be prime.

The two-prime statement (the problem): what the leads say. The abstract of [LiPa09] (arXiv v3): for every real $\beta$, "for almost all irrational $\alpha>0$ (in the sense of Lebesgue measure) $\limsup_{x\to\infty}\pi^*_{\alpha,\beta}(x)(\log x)^2/x\ge1$, where $\pi^*_{\alpha,\beta}(x)=#{p\le x:\text{both }p\text{ and }[\alpha p+\beta]\text{ are primes}}$". With $\beta=0$ this gives infinitely many primes $p$ with $[\alpha p]$ prime for almost every $\alpha$, at the heuristic order $x/(\log x)^2$ along a sequence of $x$; it says nothing about any individual $\alpha$, so the site's question, which quantifies over every irrational $\alpha>1$, is untouched. The paper's statement is known here from its abstract only. The title of [So21] names the same primes; its results are not recorded here. The citing records of [BaSh07] (52 records, scanned by title) concern Beatty primes in short intervals, in intersections of Beatty sequences and in Piatetski-Shapiro sequences, consecutive primes and bounded gaps between primes in Beatty sequences, and $k$-free values of $\lfloor\alpha p\rfloor$; none claims the two-prime statement for every $\alpha$.

Search scope. None of the routes below found a proof, disproof, preprint or proof claim for every irrational $\alpha>1$.

• The site: problem page, discussion thread and proof-claim tab; formal-conjectures at the pinned commit; the community database as accessed 2026-09-18.
• The primary sources: [Er65b] p. 210 and its reference list; [Vi48] pp. 225--226, 234--235, 243 and 246--248; [BaSh07] pp. 1, 7 and 10--11.
• arXiv API: (abs:"Beatty sequence" OR abs:"Beatty sequences") AND abs:prime sorted by date (17 records, 2007--2025; titles and journal references read; none on the two-prime question); the records of 0708.1015 (one version), 0803.1740 (three versions; abstract read) and 2005.05008 (abstract read); abs:"Erdős problem" AND (abs:951 OR abs:952 OR abs:972 OR abs:981) (no records).
• Crossref: the journal records of [BaSh07], [LiPa09] and [So21].
• Semantic Scholar: the 52 citing records of [BaSh07], scanned by title.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [LiPa09], [So21], Turán's 1937 paper, Vinogradov's Doklady notes and 1947 book. The library's 2026 card on short proofs in combinatorics and number theory, whose third theorem concerns the well-distribution of $\{\alpha p_n\}$ for Problem 997, does not bear on this question.

Remaining gaps. (1) The statement is open for every individual $\alpha$; the almost-all result is second-hand from an abstract; reopening condition: a source proving the statement for every irrational $\alpha>1$, or a disproof for some $\alpha$. (2) Source identity: the equidistribution theorem the site's argument uses is not stated in the 1948 paper, and Vinogradov's earlier work carrying it is not held; the remark above records the citation as Erdős's and the site's. (3) Proof coverage: claims checked for Theorem 5.4 and its corollaries and for the hypotheses of Theorems 1--4 of [Vi48]; nothing is proved or reviewed here, and there is no resolving proof to compile. (4) The Lean file is a statement, not a proof.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• banks_2007_prime_numbers_beatty_sequences
• banks_2007_prime_numbers_beatty_sequences / theorem_5_1
• banks_2007_prime_numbers_beatty_sequences / theorem_5_4
• erdos_1965_recent_advances_current_problems_number_theory
• vinogradov_1948_estimate_trigonometric_sums_prime_numbers
• vinogradov_1948_estimate_trigonometric_sums_prime_numbers / theorem_1
• vinogradov_1948_estimate_trigonometric_sums_prime_numbers / theorem_2
• vinogradov_1948_estimate_trigonometric_sums_prime_numbers / theorem_3
• vinogradov_1948_estimate_trigonometric_sums_prime_numbers / theorem_4