Erdoes #251 - working report - Erdős Problem a Day

Erdős problem #251 — wave8o report

Date of live-page audit and computation: 2026-07-28 UTC.

Outcome

(b+d) Verified partial result. Put

S=\sum_{n\geq 1}\frac{p_n}{2^n}.

If $S=A/B$ in lowest terms, then

\boxed{B>10^{10003}}.

This uses the named Rosser--Schoenfeld explicit upper bound for the $n$-th prime, followed by exact integer/continued-fraction computation. The least-denominator rational in the certified $N=66500$ enclosure actually has 10004 decimal digits in its denominator. A Farey-parent certificate proves its minimality, and a separate Legendre-convergent scan independently checks the displayed power-of-ten consequence.

(a) Exact structural reduction. Irrationality of $S$ is equivalent to non-eventual-periodicity of the fractional parts

\left\{\sum_{j\geq0}\frac{p_{N+j+1}-p_{N+j}}{2^j}\right\}_{N\geq1}.

Thus the exact missing lemma is now explicit: rule out eventual periodicity of these dyadically weighted future prime-gap tails. The available fixed-window prime-gap machinery located in the literature does not provide the growing-window uniformity needed for that step.

This report does not prove that $S$ is irrational.

Claim labels

(a) elementary-rigorous: a complete proof is given here.
(b) rigorous modulo the named theorem stated at the point of use.
(c) plausible or structural but unverified.
(d) computational/source observation only, with an exact reproduction

route where applicable.

Step 0: mandatory live-page audit

(d, live-page observation.) I fetched the public page through the Bright Data browser path, not datacenter curl, and also fetched its discussion thread and LaTeX-source view. The browser returned:

status OPEN;
0 claimed proofs for this problem;
Currently working on this problem: None;
Interested in collaborating: None;
This problem looks difficult: None and

This problem looks tractable: None;

both formalisation-interest/working marker rows are None;
four comments;
last page edit: 28 September 2025.

Therefore the mandatory stop condition did not trigger.

Exact current statement, verbatim from the page's LaTeX-source view

Is\[\sum \frac{p_n}{2^n}\]irrational? (Here $p_n$ is the $n$th prime.)

Source: live problem and LaTeX view, accessed 2026-07-28.

Other material listed on the live page

(d, page transcription.) The page says:

Erdős [Er58b] proved

$\sum p_n^k/n!$ irrational for every $k\geq1$.

In [Er88c] Erdős further conjectured

$\sum p_n^k/2^n$ irrational for every $k$, and conjectured that if $g_n\geq2$ and $g_n=o(p_n)$, then \[ \sum_{n=1}^{\infty}\frac{p_n}{g_1\cdots g_n} \] is irrational. It notes $g_n=p_n+1$ as evidence that some growth condition is needed.

The decimal expansion is

OEIS A098990.

The database marks the statement as formalised.
The users listed as liking the problem are ebarschkis, Prasannam,

qrdl, and jizert; none is marked as a collaborator or current worker.

All four live comments

The comments are user content and the site itself warns that they are not verified.

(d, source observation; a for the displayed identity.) TerenceTao,

17:17 on 7 October 2025, notes that summation by parts makes the problem equivalent to irrationality of $\sum_n(p_{n+1}-p_n)/2^n$. He suggests that a sufficiently quantitative and uniform prime-tuples conjecture, perhaps combined with Shannon entropy, might control the binary expansions of about $\log\log n$ consecutive gaps. The equivalence is proved independently below; the suggested route is (c).

(d, source observation.) Alfaiz, 03:28 on 15 April 2026, points to

Theorem 2 of Schlage--Puchta [ScPu11] as related. Inspection below shows why it does not directly apply to this series.

(c, unverified user claim; a for telescoping algebra only.)

Vjeko_Kovac, 11:13 on 15 April 2026, gives a claimed negative answer to the subsidiary $g_n$ conjecture. The proposed recurrence is $c_{n+1}=c_ng_n-p_n$, so, writing $Q_n=g_1\cdots g_n$, \[ \frac{p_n}{Q_n} =\frac{c_n}{Q_{n-1}}-\frac{c_{n+1}}{Q_n}. \] This telescoping identity is exact. The comment sketches, but does not fully supply on the page, the construction ensuring simultaneously that the $g_n$ are positive integers and $g_n=o(p_n)$; I do not promote that existence claim to a theorem here. It does not claim to settle the main $2^n$-denominator problem.

(d, source observation.) Nat Sothanaphan, 17:06 on 15 April 2026,

says a “standard check” found no issue with that subsidiary construction and asks for the strongest provable version. No proof details are included in this comment.

Discussion source: problem #251 discussion, accessed 2026-07-28.

Primary-source literature audit

Original sources

(d, source-identity check.) The two tracker citations exist and contain the claimed problem:

Paul Erdős, Sur certaines séries à valeur irrationnelle,

L'Enseignement Mathématique (2) 4 (1958), 93--100: archive PDF. The downloaded eight-page PDF had SHA-256 ca37134f4332be060f50cd7d6796340ff207348fe3709e1e1b26f8709c41dbf1.

Paul Erdős, *On the irrationality of certain series: problems and

results, in New Advances in Transcendence Theory* (1988), 102--109: archive PDF and chapter DOI. The downloaded eight-page archive PDF had SHA-256 b2bfc375d04b65332d6b8817633ff3968283a3f33c1f1ace366b03ac9fab8c88.

(d, primary-source reading.) Page 103 of the 1988 paper explicitly says that Erdős could not prove $\sum p_n^k/2^n$ irrational, calls even $k=1$ probably very difficult, and then states the $g_n/p_n\to0$ conjecture. This verifies that the present question is the one intended by [Er88c].

(d, bibliographic tension, not used below.) The live page describes [Er58b] as proving the factorial result for all $k$. The 1958 introduction asserts irrationality for all $k$, but says it will print only the $k=1$ proof; the theorem visible in section 3 has numerator $p_n$. Schlage--Puchta later writes that no proof for $k>1$ appeared in print. This tension concerns the factorial-denominator background, not problem #251, and none of the denominator computation below depends on resolving it.

The Schlage--Puchta comment

(d, source-identity check.) The cited paper is real: Jan-Christoph Schlage--Puchta, The irrationality of some number theoretical series, Acta Arithmetica 126 (2007), 295--303, DOI 10.4064/aa126-4-1; arXiv:1105.1451. (The journal paper is from 2007; the arXiv upload is dated 2011.) The downloaded arXiv PDF had SHA-256 8250cf8b46eb80238412570611fa0f6dca37ef8fc2f44458223e880ba05f8088.

(b, named published theorem.) Its Theorem 2 concerns the real obtained by concatenating the base-$b$ digit strings

0.f(1)f(2)f(3)\ldots,

not the fixed-place weighted sum $\sum f(n)/b^n$. Substituting $f(n)=p_n$ therefore produces the Champernowne-style concatenation of the primes, not $S$. The theorem does not settle #251.

(b, named published theorem.) The same paper's Theorem 3 proves $\mathbb Q$-linear independence of

1,\quad \sum_{n\geq1}\frac1{n!},\quad \sum_{n\geq1}\frac{p_n}{n!},\quad \sum_{n\geq1}\frac{p_n^2}{n!},\ldots .

Its Lemma 4 also says that a fixed nonzero polynomial in a fixed number of consecutive prime gaps is nonzero for almost every index. This fixed-window result is relevant context, but the wall below requires control uniform in a window whose length grows.

Search result and honest miss

(d, search observation.) Exact-phrase web searches for the displayed series, searches by the original title, the OEIS references, Crossref, and the arXiv API located the tracker, OEIS, the two Erdős sources, and the Schlage--Puchta paper, but no primary paper claiming a resolution of the main $2^n$-denominator question. In particular, the arXiv API query all:"nth prime" AND all:irrationality returned totalResults=0; a broader query for “number theoretical series” and irrationality returned arXiv:1105.1451 and 1105.1452, of which only the former involves the factorial prime series. (c) This is not a proof that no other literature exists; it is the precise scope and outcome of the search.

Exact structural reduction

Let

\delta_n=p_{n+1}-p_n,\qquad S_N=\sum_{n=1}^{N}\frac{p_n}{2^n}.

Summation by parts

(a, algebra; b only for passage to the convergent limit.)

\begin{aligned} \sum_{n\geq1}\frac{\delta_n}{2^n} &=\sum_{n\geq1}\frac{p_{n+1}}{2^n} -\sum_{n\geq1}\frac{p_n}{2^n}\\ &=2\sum_{m\geq2}\frac{p_m}{2^m}-S\\ &=2(S-p_1/2)-S=S-2, \end{aligned}

because $p_1=2$. Hence

\boxed{S=2+\sum_{n\geq1}\frac{\delta_n}{2^n}}. \tag{1}

The checker independently verifies the exact finite identity, including its boundary term.

The future-gap orbit

Define the convergent gap tail

G_N=\sum_{j\geq0}\frac{\delta_{N+j}}{2^j}.

Also put

X_N=2^N(S-S_N)=\sum_{j\geq1}\frac{p_{N+j}}{2^j}.

(a). Since

p_{N+j}=p_N+\sum_{i=0}^{j-1}\delta_{N+i} \quad\text{and}\quad \sum_{j\geq i+1}2^{-j}=2^{-i},

interchanging these absolutely convergent nonnegative sums gives

\boxed{X_N=p_N+G_N}. \tag{2}

Since $2^N S_N$ and $p_N$ are integers,

\boxed{\{2^N S\}=\{G_N\}}. \tag{3}

(a, exact equivalence). A real number $x$ is rational if and only if the sequence $\{2^N x\}$ is eventually periodic. For rational $x=a/b$, the residues $2^Na\bmod b$ form an eventually periodic finite-state sequence. Conversely, if $\{2^{N+m}x\}=\{2^Nx\}$ for some $N,m\geq1$, then

(2^{N+m}-2^N)x\in\mathbb Z,

so $x\in\mathbb Q$. Combining this with (3) yields

\boxed{S\in\mathbb Q \iff (\{G_N\})_{N\geq1}\text{ is eventually periodic}.} \tag{4}

(a, necessary carry condition). The tails obey

G_{N+1}=2(G_N-\delta_N). \tag{5}

If the fractional parts in (4) eventually have period $m$, then $G_{N+m}-G_N\in\mathbb Z$ for all sufficiently large $N$. Splitting the first $m$ terms from $G_N$ gives

2^mG_N= \sum_{j=0}^{m-1}2^{m-j}\delta_{N+j}+G_{N+m},

and therefore

\boxed{(2^m-1)G_N\in\mathbb Z \quad\text{for all sufficiently large }N.} \tag{6}

Equations (4)--(6) expose the binary carries that a proof must control.

A rigorous finite denominator bound

Rational enclosure

Let

A_N=\sum_{n=1}^{N}p_n2^{N-n}\in\mathbb Z, \qquad S_N=\frac{A_N}{2^N}.

(b, Rosser--Schoenfeld.) The corollary to Theorem 3 of J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64--94, DOI 10.1215/ijm/1255631807, gives

p_k<k(\log k+\log\log k)\qquad(k\geq6).

For $k\geq6$, this implies $p_k<2k^2$. The elementary identity

\sum_{k=N+1}^{\infty}\frac{k^2}{2^k} =\frac{N^2+4N+6}{2^N}

then gives

\frac{p_{N+1}}{2^{N+1}} <S-S_N <\frac{2(N^2+4N+6)}{2^N}. \tag{7}

The checker verifies the polynomial-tail identity by its exact recurrence.

Put $C_N=2(N^2+4N+6)$. Equation (7) places $S$ strictly inside the closed rational interval

I_N= \left[ \frac{2A_N+p_{N+1}}{2^{N+1}}, \frac{A_N+C_N}{2^N} \right]. \tag{8}

Why the least denominator is exactly certified

(a, Farey lemma). Suppose $a_1/b_1<a_2/b_2$ are reduced and $a_2b_1-a_1b_2=1$. If a reduced $u/v$ lies strictly between them, then the two positive integer cross-differences give

\frac1{b_1b_2} =\frac{a_2}{b_2}-\frac{a_1}{b_1} \geq\frac1{vb_1}+\frac1{vb_2},

and hence $v\geq b_1+b_2$.

For every row below, the program obtains $a/b\in I_N$, constructs its two Farey parents $a_1/b_1,a_2/b_2$, and checks using exact integers that

a=a_1+a_2,\quad b=b_1+b_2,\quad a_2b_1-a_1b_2=1,

and

\frac{a_1}{b_1}<\inf I_N\leq\frac ab\leq\sup I_N <\frac{a_2}{b_2}.

The Farey lemma therefore certifies that $b$, denoted $q_{\min}(I_N)$, is the least reduced denominator of any rational in $I_N$. This certificate does not depend on trusting the search routine that found $a/b$.

Exact computation

(d, exact integer computation; b for its application to $S$.) The compact denominator fingerprint is first 20 digits … last 20 digits; the final column is the first 16 hexadecimal characters of SHA-256 of the full decimal denominator. Small values are shown in full. The standalone checker recomputes every full integer.

$N$	$p_{N+1}$	CF steps	digits of $q_{\min}(I_N)$	compact $q_{\min}(I_N)$	SHA-256 prefix
25	101	6	3	292	`6db6eb4af1e18ab8`
50	233	12	6	361693	`29b5b1f22f92ccb8`
100	547	28	14	15300103749941	`a48020aabee7b455`
250	1597	69	36	25679262580425917790…77908635532163719998	`489927ff97a430e2`
500	3581	139	73	33968216970513425812…45979616280486873109	`0a7bc893535a3d1c`
1000	7927	295	148	85007982564349604827…71429521793445100321	`6383a6f913b86cb1`
5000	48619	1464	749	88660973688832503488…73842669067232053957	`b695e4f22a1ebd83`
10000	104743	2908	1501	34044923053342575976…35962258427406986274	`641f90ece1d0e160`
25000	287137	7300	3759	96466203223523242427…27687734759718289603	`c7c3ad41ced0c74d`
66500	834571	19336	10004	71364913553098073984…37645606155679084776	`7d0683df3dcc22ad`

The final row and (8) imply

B\geq q_{\min}(I_{66500})>10^{10003}

if $S=A/B$ is in lowest terms.

Independent check of the headline exponent

(b for Legendre's criterion and the tail bound; d for enumeration.) Let $x=A_N/2^N$, $Q=10^{10003}$, and $N=66500$. The program checks the exact integer inequality

\[ 2C_NQ^2<2^N, \]

so (7) gives

0<S-x<\frac1{2Q^2}.

If $S=A/B$ with $B\leq Q$, Legendre's continued-fraction criterion would make $A/B$ a convergent of the exact rational $x$. Independently of the Farey-parent calculation, the checker enumerated all 19336 convergents of $x$ having denominator at most $Q$ and verified by exact cross-multiplication that none lies in the possible tail interval. This is a second verification of $B>10^{10003}$.

Reproduction

The standalone checker is erdos251_wave8o_reverify.py. It uses only the Python standard library.

Run:

cd /home/exedev/MathDyad
python runs/erdos251_wave8o_reverify.py

(d, measured on this VM.) Python 3.12.3 on Linux x86-64 produced:

prime cross-check: 66501 primes, last=834571, sieve=0.118s, trial=0.620s
...
66500   834571     19336    10004     71364913553098073984  37645606155679084776  7d0683df3dcc22ad
independent Legendre check: 19336 convergents with denominator <= 10^10003; none lies in the tail interval
VERIFIED: if S is rational in lowest terms A/B, then B > 10^10003.
total runtime: 69.724s

Peak resident memory reported by /usr/bin/time was 23536 KiB. The script:

generates the first 66501 primes by Eratosthenes;
independently regenerates and compares them by trial division;
verifies finite forms of (1) and (2);
checks the square-tail recurrence;
exhaustively self-tests the simplest-fraction routine on a small rational

grid against brute force;

constructs and verifies a Farey-parent certificate for every table row;
performs the independent Legendre-convergent scan.

The only non-finite input to the certificate is the explicitly named Rosser--Schoenfeld prime bound.

Exact wall

(a). No finite denominator exclusion can prove irrationality: a rational number may have an arbitrarily large denominator. Increasing $N$ in (8) only raises a finite lower bound for $B$.

(a). By (4), the exact theorem still needed is:

For every $m\geq1$ and every $N_0$, there is an $N\geq N_0$ such that $G_{N+m}-G_N\notin\mathbb Z$.

Equivalently, one must rule out eventual periodicity of $\{G_N\}$. Equation (5) shows why ordinary information about one gap at a time does not suffice: binary carries propagate from the entire future tail.

(c, quantitative diagnosis.) On the usual scale $\delta_n\asymp\log n$, truncating $G_N$ with $O(1)$ absolute error requires roughly $\log_2\log N$ future gaps. This explains the growing-window length in Tao's page comment. To turn that heuristic into a proof one needs a uniform result controlling the relevant joint residues and binary carries for consecutive prime gaps in a window $r=r(N)\to\infty$, at least around $r\asymp\log\log N$.

(b+c). Schlage--Puchta's Lemma 4 rigorously excludes a fixed nonzero polynomial relation among a fixed number of consecutive gaps for almost all $N$. It does not supply estimates uniform as the number of gaps grows, nor does it encode the unbounded carry in $G_N$. The missing uniformity/finiteness step is exactly why citing that lemma or Theorem 2 does not close #251.

(c, cost estimate, not run.) Merely extending the enclosure to $N=10^6$ should yield a denominator certificate of roughly 150000 decimal digits, but extrapolating the measured big-integer continued-fraction work puts the pure-Python run at approximately 4--8 single-core hours. It would still be only a finite denominator bound and would not address the missing growing-window prime-gap theorem, so that heavier computation was not run.

PARTIAL: The main irrationality question remains open; exact reduction (4) isolates eventual periodicity of weighted prime-gap tails, and a reproducible Farey/continued-fraction certificate proves that any rational value must have reduced denominator greater than 10^10003.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger