Eight Erdős problems

Read the mathematics. Find what comes next.

This site collects papers about eight mathematical questions, the results proved so far and the questions that remain open. Start with a problem below or its short paper. The longer papers describe the investigation, including attempts that failed. The GitHub repository contains the papers, proof files and programs used for calculations.

Open the map of results Read the starting guide

689 results from the sixteen problem papers, linked to formal statements, proof files and recorded tests of those proofs.

S = \sum_{n \geq 2} \frac{1}{n! - 1}

#68 The factorial-denominator series

Is the series $\sum_{n \geq 2} 1 / (n! - 1)$ irrational? The question is open. Read the short paper

lim_{n \to \infty} \frac{a_{n}}{a_{n - 1}^{2}} = 1 and \sum \frac{1}{a_{n}} \in Q

#243 Reciprocal-tail rigidity near the Sylvester recurrence

Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually? The question is open. Read the short paper

S = \sum_{n \geq 1} \frac{φ (n)}{2^{n}}

#249 The binary totient series

Is the binary Lambert series $\sum φ (n) / 2^{n}$ irrational? The question is open. Read the short paper

Π = \sum_{n \geq 0} \frac{p_{n}}{2^{n + 1}}

#251 The prime-gap dyadic series

Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational? The question is open. Read the short paper

X_{A} (b) = \sum_{a \in A} \frac{1}{b^{a} - 1}

#257 Reciprocal sums over infinite exponent supports

Is the sum of $1 / (2^{n} - 1)$ over every infinite set of positive exponents irrational? The question is open. Read the short paper

R_{P} = \sum_{u \in S_{P}} \frac{1}{L (u)}

#269 Three-prime running least common multiples

For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? The question is open. Read the short paper

Ω_{f} = {z : | f (z) | < 1}

#1041 Short connections inside polynomial lemniscates

Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate? Status: formal statement refuted. Read the short paper

F (t) = \sum_{n \geq 1} (t^{n} - 1)^{- 1}

#1049 Lambert-type series at rational bases

For which rational bases is the corresponding series irrational? The question is open. Read the short paper

Problems

Each problem page separates statements with Lean proofs from unanswered questions. Lean is software that verifies formal proofs against their statements and assumptions.

#68The factorial-denominator series
Is the series $\sum_{n \geq 2} 1 / (n! - 1)$ irrational?

Short paper Long paper Map of results
#243Reciprocal-tail rigidity near the Sylvester recurrence
Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?

Short paper Long paper Map of results
#249The binary totient series
Is the binary Lambert series $\sum φ (n) / 2^{n}$ irrational?

Short paper Long paper Map of results
#251The prime-gap dyadic series
Is the dyadic series of consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?

Short paper Long paper Map of results
#257Reciprocal sums over infinite exponent supports
Is the sum of $1 / (2^{n} - 1)$ over every infinite set of positive exponents irrational?

Proved here: if a set of positive integers satisfies the theorem’s finite prime-weighted summability condition in base 2, every infinite subset defines an irrational series of terms 1/(bⁿ − 1) at every integer base b ≥ 2, where n ranges over that subset. Read the theorem and its hypotheses. The original universal question remains open; novelty is unassessed.

Short paper Long paper Expository paper See the 189/388 calculation Map of results
#269Three-prime running least common multiples
For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational?

This library treats the three-prime case.

Short paper Long paper Map of results
#1041Short connections inside polynomial lemniscates
Must two roots of a monic polynomial in the open unit disc be joined by a sub-two-length curve inside its unit lemniscate?

Ani’s degree-seven example refutes the exact Formal Conjectures statement; correspondence with the 1958 wording awaits human review.

Short paper Long paper Map of results
#1049Lambert-type series at rational bases
For which rational bases is the corresponding series irrational?

The first resistant explicit base is three halves.

Short paper Long paper Map of results

Reading the eight together

Read the papers and Lean corpus together to find connections, reusable constructions and new questions beyond the original targets. This paper develops that work, keeping each result’s proof and remaining limits visible.

Reading Eight Erdős Problems Together
When do sparse integer digits fill intervals of values, when do they force irrationality or transcendence, and what does that show about the methods used across the Erdős problems?
Read the paper PDFOpen the PDF, 38 pages

Map of results

Every result in the papers, drawn with its problem

689 results from the problem papers, in a ring round the problems and their claims. The colours distinguish results stated in Lean, a language for computer-verified proofs, from results whose proofs Comparator has also tested against separate theorem statements. Hover to read a title; select a result to read its statement, assumptions and proof records.

Open the full map

The papers

Every programme above opens its own papers: sixteen problem papers, a short paper and a longer record for each, and one expository paper. The papers catalogue lists all 22 current papers with their lengths and PDFs, and keeps the retired manuscripts apart.

Open the papers catalogue

Method and the research system

Writing a Good Mathematical PaperExposition method, 2 pages PDF
Writing Mathematics from the Literature and Reviewed RevisionsExposition method, 21 pages PDF
A Repository-Based System for Research and PublicationSystems paper, 25 pages PDF
Plectis: What a Stranger Can CheckSystems paper, 22 pages PDF

Reading and checking the work

Technical reference and history

Develop what comes next

Explore with your AI. One reading file, no clone: an introduction, a short research instruction and the opening of each paper.
Work in the repository. Download a working copy, name a question, or ask your coding agent to read the papers and proof files and propose further work.
Choose an unanswered question. One GitHub issue lists an unanswered research question for each of the eight problems.

You can contribute an insight before writing a formal proof. Will can work with you to develop and formalise it, with credit for the insight and the subsequent work kept distinct. The whole collection is also material for new mathematics: connections, reusable methods and questions across problems.

Will Cook built and directs this AI-assisted research environment and maintains its public release. AI agents did most of the research and drafting. The papers name earlier sources and contributors to particular results, including ani’s #1041 construction. For a result, cite its paper and relevant earlier sources; for software, use the release citation and give the commit used. Contact Will with questions or corrections.

About this edition

8 problems, 22 paper records in the source snapshot, 18 guides and 2,907 mapped objects.

Papers and maps are generated together from one source snapshot. GitHub links open the current source. Edition b0ebeb9add157584.