Problem 251

../

claims/: The 2 claim pages of Problem 251, one per claimant's result; the problem's standing derives from them.

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Statement. Is

$$\sum \frac{p_n}{2^n}$$

irrational? (Here $p_n$ is the $n$th prime.)

Status. Open, as the site's label (OPEN) also has it. No unconditional proof or disproof is known. Two claimed conditional proofs, both under Kuperberg's uniform Hardy–Littlewood prime-tuples conjecture, have claim pages, Land and Ringer; neither is refereed or independently reviewed, and both would leave the problem open even if accepted.

Source. erdosproblems.com/251, accessed 2026-09-17 (page last edited 28 September 2025). Cite as: T. F. Bloom, Erdős Problem #251, https://www.erdosproblems.com/251, accessed 2026-09-17.

References.

• [Er58b] Erdős, P., Sur certaines séries à valeur irrationnelle. Enseign. Math. (2) 4 (1958), 93--100.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève, Geneva, 1980; p. 62.
• [Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986), Cambridge Univ. Press, Cambridge, 1988, 102--109; p. 103.
• [Er61] Erdős, P., Számelméleti megjegyzések, I. (Remarks on number theory, I.). Mat. Lapok 12 (1961), 10--17.
• [ErSt74] Erdős, P. and Straus, E. G., On the irrationality of certain series. Pacific J. Math. 55 (1974), 85--92.
• [ErPo78] Erdős, P. and Pomerance, C., On the largest prime factors of $n$ and $n+1$. Aequationes Math. 17 (1978), 311--321.
• [HaTi04] Hančl, J. and Tijdeman, R., On the irrationality of Cantor series. J. Reine Angew. Math. 571 (2004), 145--158.
• [ScPu07] Schlage-Puchta, J.-C., The irrationality of some number theoretical series. Acta Arith. 126 (2007), no. 4, 295--303; arXiv:1105.1451 (the site's key [ScPu11]).
• [Po12] Pollack, P., The average least quadratic nonresidue modulo $m$ and other variations on a theme of Erdős. J. Number Theory 132 (2012), no. 6, 1185--1202.
• [Ku23] Kuperberg, V., Sums of singular series with large sets and the tail of the distribution of primes. Q. J. Math. 74 (2023), no. 4, 1457--1479; arXiv:2210.09775.
• [Ta23] Tao, T., The convergence of an alternating series of Erdős, assuming the Hardy–Littlewood prime tuples conjecture. arXiv:2308.07205 (2023); Comm. Amer. Math. Soc. 4 (2024), 80--96.
• OEIS A098990, decimal expansion of $\sum_{n\ge1}p_n/2^n$.

Formalization. The statement is recorded in formal-conjectures as answer(sorry) ↔ Irrational (∑' n : ℕ, (Nat.nth Nat.Prime n) / (2 ^ n)) with the tag category research open and a sorry proof (the file's revision as of 2026-09-17, the commit of 2026-07-16 that the link carries). Since Nat.nth Nat.Prime is zero-based, that real number is $\sum_{n\ge0}p_{n+1}/2^n=2S$, twice the site's constant $S$; irrationality is unaffected, but the formal object is not the site's number. Two author repositories hold conditional Lean statements with Kuperberg's conjecture as an explicit hypothesis (see Progress); neither is part of this repository's accepted Lean closure, and no statement of either has been compared with its manuscript.

Current assessment

The site's formulation (last edited 28 September 2025) asks whether $S=\sum_{n\ge1}p_n/2^n$ is irrational, with $p_1=2$, so $S=2/2+3/4+5/8+\cdots=3.67464396601\ldots$ (OEIS A098990). The frontmatter status open concerns this question. The status search covered the primary sources (the 1958, 1980 and 1988 passages), the arXiv API, the catalog's problem page, discussion thread, proof-claims page and history page, the community database and its "AI contributions" wiki, and web search; zbMATH, MathSciNet and Google Scholar were not searched. No unconditional proof or disproof was found. A failure to find a proof does not by itself establish openness, so the scope is recorded and the outstanding claims are listed.

Origin and wording. Erdős posed the question in 1958 (p. 94), after proving $\sum p_n/n!$ irrational: "je n'ai pas réussi à démontrer l'irrationnalité de la somme des séries $\sum_{n=1}^{\infty}1/(p_n\,n!)$, $\sum_{n=1}^{\infty}p_n/2^n$ et $\sum_{n=1}^{\infty}1/(p_n\,2^n)$"; no later work on the two companion series was found. The 1980 monograph (p. 62) says "the irrationality of $\sum_np_n/2^n$ is probably hopeless." The 1988 survey (p. 103) writes: "I could not prove that $\sum p_n^k/2^n$ is irrational for every $k$. This is probably very difficult already for $k=1$. It seems reasonable to expect that if $g_n\ge2$, $g_n/p_n\to0$ then $\sum_{n=1}^{\infty}p_n/g_1\ldots g_n$ (2) is irrational, but I can prove the irrationality of (2) only under much more restrictive conditions; $g_n=p_n+1$ shows that some growth condition is needed for the irrationality of (2)." (problem_p103 carries the passage.) The site's word "conjectures" for these two statements is stronger than Erdős's "could not prove" and "seems reasonable to expect"; the main question ($k=1$) is the same in every source. The status judges the Statement; the $k\ge2$ power series and the variable-denominator statement are auxiliary questions treated below.

The site remark on $\sum p_n^k/n!$. The site says Erdős [Er58b] proved $\sum p_n^k/n!$ irrational for every $k\ge1$. The 1958 paper proves $k=1$ (Section 2) and says the proof for $k>1$ is too complicated to include; the 1980 book repeats the all-$k$ attribution. The first published proof for $k\ge2$ is Schlage-Puchta's Theorem 3 [ScPu07] (theorem_3): $1,S_0,S_1,S_2,\dots$ are $\mathbb Q$-linearly independent, where $S_k=\sum p_n^k/n!$; Schlage-Puchta's paper says (p. 2) "it appears that, for $k>1$, no proof has appeared in print." This corrects the background, not the problem.

Reformulation. With $g_m=p_{m+1}-p_m$, $S=2+\sum_{m\ge1}g_m/2^m$, a two-line summation by parts proved on reformulation_gap_series; the question is the irrationality of the dyadic prime-gap series, and every 2026 manuscript works in that form.

The auxiliary variable-denominator statement: a claimed counterexample. Theorem 1 of the two-page note bylined "ChatGPT 5.4 Pro (orchestrated by Vjeko Kovač)" (2026-04-15) constructs integers $g_n\ge2$ with $g_n=o(p_n)$ and $\sum p_n/(g_1\cdots g_n)=1$. The argument is elementary; this compilation carries a proof sketch as author-recorded, and no independent review is filed. The note is unrefereed and AI-generated by its byline, the check reported on the discussion thread was AI-run, the community wiki lists the note as "Solution to variant problem", and the site's text is unchanged. Kovač's own comment notes that Erdős may have intended nondecreasing $g_n$; the construction is not monotone, so the monotone-denominator theorems below are untouched. The construction concerns one particular sequence, not the constant sequence $g_n=2$ of the problem. This page therefore records a claimed counterexample to the auxiliary statement as a body note, not a disproof and not a claim page, since it settles no instance of the question; the problem's status is unaffected either way.

Conditional claims (2026). Both assume Kuperberg's Conjecture 1.3 (conjecture_1_3), a Hardy–Littlewood prime-tuples conjecture with one power-saving error uniform over tuples of size up to $(\log\log x)^3$ in $[0,(\log x)^2]$, for which no unconditional support at that uniformity exists. Both are unrefereed manuscripts hosted on GitHub and not found on arXiv; both report author-run Lean builds and axiom audits of conditional statements and disclaim independent reproduction; no independent review of either is filed or was found; acceptance of either would establish an implication, not the irrationality, and would leave the status open.

• J. Land, research draft dated 5 September 2026, announced on the discussion thread on 2026-09-06 (claim page Land): Theorem 2, Conjecture 1 (Kuperberg's conjecture in large-$x$ form) implies $S\notin\mathbb Q$, through weighted prime-gap tails $G_n$ with $G_n>6$ and $G_n\to6$. The paper states it was "prepared with the assistance of the AI systems gpt-6-astra, fable 5.1, and gemini-3.8-flash" and that "no proof-assistant verification is claimed"; the repository's terminal theorem Erdos251.erdos251_conditional (hK : UniformHardyLittlewoodConjecture) : Irrational Erdos251.realSeries, with realSeries = ∑' n, (p n : ℝ) / 2^(n+1) and zero-based p, encodes $S$ exactly; the repository's README reports an author-run build and axiom audit. No proof claim is registered on the site. Claimed, unreviewed.
• S. Ringer, manuscript dated 11 September 2026, the site's single registered proof claim (submitted 2026-09-13 and labeled partial, with the AI systems GPT 6 Astra and Fable 5.1 named in its title; claim page Ringer): Corollary 1.2, under the positive-comparison hypothesis (19) with $\kappa\ge1/\log2$, which the averaged condition $(\mathrm{AHL}_\kappa)$ implies and Kuperberg's conjecture implies in turn (Section 5.4), $\sum p_n2^{-n}$ is normal to base $2$, hence irrational; Theorem 1.1 classifies periodic polynomial gap series. The paper's provenance paragraph reads "GPT 6 Astra led the mathematical development, and Fable 5.1 acted as a sparring partner." The "partial" label matches the paper's own scope: a conditional result, with the repository README saying the original problem remains open unconditionally. Claimed, unreviewed.

The two hypotheses are different specializations of the same conjecture; Ringer states that no implication between them is claimed.

Research context, not progress. W. Cook's discussion comment (2026-09-11) and note "A countermodel for growth-and-parity arguments on the prime-gap dyadic series" (paper/251/ of wcook04/plectis-erdos) construct a sparse perturbation of the prime gaps whose dyadic sum is rational while the growth scale, every fixed eventual congruence and the short-block statistics are kept; the author presents it as a countermodel to weaker hypotheses rather than a solution, says that AI tools contributed substantially to it, and says that the notes have had no independent mathematical review. It is a statement about integer sequences, not primes, and is not filed as a source.

Adjacent theorems (historical; not progress on $S$).

• Erdős 1958, Section 3 (pp. 96--99; the theorem on pp. 96--97, its proof on pp. 97--99): for integers $1<q_1\le q_2\le\cdots$ satisfying the growth hypothesis (5), printed as $q_n>o(n/\log^kn)$ for some $k>0$, the sum $\sum p_n/(q_1\cdots q_n)$ is rational if and only if $q_n=qp_n+1$ for a fixed integer $q\ge1$ and all $n\ge n_0$; the closing remark on p. 99 says the case $q_n=2$ "m'échappe entièrement." Exact statement: theorem_section_3; the p. 94 list of unresolved series: remark_p94.
• Hančl–Tijdeman 2004 [HaTi04], Theorem 5.1 (preprint p. 8): for a monotonic sequence of positive integers $a_n$ with $p_n=o(a_n^2)$, $\sum p_n/(a_1\cdots a_n)$ is rational if and only if $p_n/(a_n-1)$ is constant for $n\ge n_0$; their remark on p. 9 shows by example that the monotonicity cannot be dropped. Example 3.1 (p. 4): $\sum p_n^k/2^{p_n}$ is irrational for every integer $k>0$; the paper attributes the case $k=1$ to [ErGr80], p. 62, which names no prime series of this shape but says that $\sum a_n/2^{a_n}$ "is known to be irrational under the stronger hypothesis that $a_n>cn\sqrt{\log n\log\log n}$", a hypothesis $a_n=p_n$ satisfies. The statements and page numbers follow the authors' preprint. Card: hancl_2004_irrationality_cantor_series.
• Erdős–Straus 1974 [ErSt74] gives rationality criteria for $\sum b_n/(a_1\cdots a_n)$ with monotone denominators, including a reproof of the $k=1$ factorial theorem; cited by statement. Card: erdos_1974_irrationality_certain_series.
• Erdős–Pomerance 1978 [ErPo78], p. 320, quoted beside the problem in [ErGr80]: with $\varepsilon_n=1$ if $P(n)>P(n+1)$ and $0$ if $P(n)<P(n+1)$ ([ErGr80] states the reverse convention, whose series is a rational number minus this one), $P(n)$ the largest prime factor, $\sum_{n\ge2}\varepsilon_n/2^n$ is irrational, a $0/1$ series without carries; for $S$ the carries are the difficulty. Card: erdos_1978_largest_prime_factors.
• The constant. Erdős 1961 [Er61], equation (3): $\sum_{p<x}n_2(p)=(1+o(1))\,(x/\log x)\sum_{k\ge1}p_k/2^k$, so $S$ is the mean of the least quadratic nonresidue over primes (the theorem of Problem 980; English statement in [Po12], eq. (1.1); OEIS A098990). The identity says nothing about irrationality.

Proof coverage and review. No proof on or linked from this page is independently reviewed. The Kovač theorem carries an author-recorded proof sketch; the reformulation is a checked two-line identity; the conditional claims carry statements with proof pointers; the historical theorems are cited by statement.

Progress

Dated record of work on the exact question; none changes the status.

• 2025-10-07: T. Tao, discussion comment, the gap-series reformulation (reformulation_gap_series) and the suggestion that a quantitative prime tuples conjecture, uniform enough to control the binary digits of about $\log\log n$ consecutive gaps, might resolve the problem.
• 2026-09-05/06: J. Land, Theorem 2: Kuperberg's Conjecture 1.3 (large-$x$ form) implies $S\notin\mathbb Q$. Claimed, unrefereed, AI-assisted by the paper's own statement; author-run Lean build of the conditional statement; no independent review. Claim page: Land.
• 2026-09-07/13: S. Ringer, Corollary 1.2: under the positive-comparison hypothesis (19), implied by $(\mathrm{AHL}_\kappa)$ and so by Kuperberg's conjecture, $\sum p_n2^{-n}$ is normal to base $2$, hence irrational; announced on the thread on 2026-09-07 (the repository's first commit is dated 2026-09-10; the card digests the text at the commit of 2026-09-13) and registered on the site on 2026-09-13 as a partial proof claim naming the AI systems GPT 6 Astra and Fable 5.1; author-run Lean build; no independent review. Claim page: Ringer.

The dated site record (page, discussion, proof claim, history) is filed as bloom_2026_erdos_problem_251_discussion.

Known Results

Auxiliary statement and adjacent theorems, each with its source:

• Kovač-orchestrated note, Theorem 1 (2026-04-15): a claimed counterexample to the $g_n$ statement of [Er88c], p. 103; author-recorded proof sketch; no independent review. A body note, not a claim, since it settles no instance of the question.
• Erdős 1958: $\sum p_n/n!$ irrational (main_theorem; $k=1$ proved, $k>1$ asserted); the Section 3 theorem for nondecreasing denominators (theorem_section_3); the p. 94 list of the three unresolved series, the source of this problem (remark_p94).
• Schlage-Puchta 2007, Theorem 3: $1,S_0,S_1,\dots$ are $\mathbb Q$-linearly independent, the $k\ge2$ case of the site remark in print.
• Erdős 1988, p. 103, and Erdős–Graham 1980, p. 62: the statements quoted in the assessment.
• Hančl–Tijdeman 2004, Theorem 5.1 and Example 3.1, and Erdős–Straus 1974: the monotone-denominator criteria cited by statement above.
• Kuperberg's Conjecture 1.3: the hypothesis of both conditional claims; a conjecture, not a result.
• Erdős 1961, equation (3): $S$ as the mean least quadratic nonresidue over primes.

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.

• erdos_1978_largest_prime_factors
• erdos_1978_largest_prime_factors / theorem_p320
• erdos_1961_szamelmeleti_megjegyzesek
• erdos_1961_szamelmeleti_megjegyzesek / equation_3
• bloom_2026_erdos_problem_251_discussion
• bloom_2026_erdos_problem_251_discussion / bloom_2026_erdos_problem_251_discussion
• bloom_2026_erdos_problem_251_discussion / reformulation_gap_series
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french / density_lemma
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french / main_theorem
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french / proposition_p95
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french / remark_p94
• erdos_1958_sur_certaines_series_valeur_irrationnelle_french / theorem_section_3
• erdos_1974_irrationality_certain_series
• erdos_1974_irrationality_certain_series / corollary_2_10
• erdos_1974_irrationality_certain_series / theorem_3_1
• erdos_1988_irrationality_certain_series_problems_results
• erdos_1988_irrationality_certain_series_problems_results / problem_p103
• hancl_2004_irrationality_cantor_series
• hancl_2004_irrationality_cantor_series / algorithm_3_1
• hancl_2004_irrationality_cantor_series / corollary_4_2
• hancl_2004_irrationality_cantor_series / example_3_1
• hancl_2004_irrationality_cantor_series / theorem_5_1
• hancl_2004_irrationality_cantor_series / theorem_6_1
• hancl_2005_irrationality_factorial_series
• hancl_2005_irrationality_factorial_series / theorem_3_4
• hancl_2010_irrationality_factorial_series_ii
• hancl_2010_irrationality_factorial_series_ii / corollary_3_2
• kovac_2026_erdos_problem_251
• kovac_2026_erdos_problem_251 / theorem_1
• land_2026_conditional_proof_irrationality_prime_series
• land_2026_conditional_proof_irrationality_prime_series / theorem_2
• ringer_2026_local_gap_statistics_telescoping_normality
• ringer_2026_local_gap_statistics_telescoping_normality / corollary_1_2
• ringer_2026_local_gap_statistics_telescoping_normality / theorem_1_1
• schlagepuchta_2011_irrationality_number_theoretical_series
• schlagepuchta_2011_irrationality_number_theoretical_series / lemma_4
• schlagepuchta_2011_irrationality_number_theoretical_series / theorem_2
• schlagepuchta_2011_irrationality_number_theoretical_series / theorem_3
• tijdeman_2002_rationality_cantor_ahmes_series
• tijdeman_2002_rationality_cantor_ahmes_series / theorem_3_1
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• kuperberg_2023_sums_singular_series_large_sets_tail
• kuperberg_2023_sums_singular_series_large_sets_tail / conjecture_1_3