Reading Eight Erdős Problems Together

Will Cook

Cross-problem paper

Reading Eight Erdős Problems Together

Eight Erdős problems read together By Will Cook 22 September 2026 Authorship, AI use and citation

PDFOpen the PDF, 38 pages TeXLaTeX source Source on GitHub

Précis. One account of the mathematics developed across the programmes: an exact tail-capacity criterion under eventual congruences, its sharp factorial support-gap threshold, rational and irrational Lambert subsums across bases, and the limits of finite greedy tests and other irrationality methods. Full arguments, exact counterexamples, unsuccessful routes and attribution are retained together. The principal proofs are ordinary mathematics; cited Lean ingredients have their own stated scope. Historical novelty and independent expert review are not established.

This paper owns the synthesis exposition and ordinary proofs across the covered problems: capacity and congruence constructions, Lambert subsums, rational-point counts, method limits and their research record.

It is not authority for a solution to any original Erdős target, historical novelty, independent expert review, or a full Lean proof of the analytic capacity criterion or Lambert-chain theorem.

Start here, by the editorial route

In this paper

A common-divisor test for irrationality

The distinction between small and large allowances also appears directly in rationality. Let $b_{n} \geq 2$ be integers, put $Q_{0} = 1$ and $Q_{n} = b_{1} \dots b_{n}$ , and consider a Cantor series.

Theorem 3.6. Suppose $0 \leq a_{n} \leq A$ and $e_{n} \geq 0$ are integers, $e_{n} \leq C b_{n}$ eventually, and $a_{n} + e_{n}$ is nonzero infinitely often. If

\underset{n \to \infty}{lim sup} gcd (b_{n}, e_{n}) = \infty,

then $\sum_{n \geq 1} (a_{n} + e_{n}) / Q_{n}$ is irrational.

Proof. For large $n$ , the scaled tail satisfies

0 < Q_{n} \sum_{k > n} \frac{a_{k} + e_{k}}{Q_{k}} \leq A + 2 C .

Indeed, the bounded $a_{k}$ contribute at most $A \sum_{j \geq 1} 2^{- j} = A$ , and $e_{k} \leq C b_{k}$ contributes at most $C \sum_{j \geq 1} 2^{- (j - 1)} = 2 C$ . If the total were $p / D$ , the numbers

T_{n} = D Q_{n} (\frac{p}{D} - \sum_{k \leq n} \frac{a_{k} + e_{k}}{Q_{k}})

would be positive integers eventually bounded by $D (A + 2 C)$ . No hypothesis that $D ∣ Q_{n}$ is needed: the extra factor $D$ clears it. The recurrence

b_{n} T_{n - 1} = D (a_{n} + e_{n}) + T_{n}

implies $gcd (b_{n}, e_{n}) ∣ D a_{n} + T_{n}$ . The integer on the right is positive and eventually at most $D (2 A + 2 C)$ , contradicting the unbounded gcd. ◻

For factorial denominators, $b_{n} = n$ , and eventual divisibility of $e_{n}$ by each fixed integer makes $gcd (n, e_{n})$ arbitrarily large arbitrarily late: choose a late multiple of that integer. Hence bounded nonnegative digits that are nonzero infinitely often cannot be rationalised by nonnegative $O (n)$ corrections satisfying (2). Conversely any allowance $F (n)$ with $F (n) / n \to \infty$ permits interval filling under (4): apply the positive construction with $r_{n} = n$ , replacing $n^{c}$ by $F (n)$ .

The arithmetic hypothesis cannot be replaced by $b_{n} \to \infty$ , even if both congruences (4) are retained. Set $a_{n} = 1$ , $e_{1} = 2$ , and $e_{n} = (n + 1)! - n!$ for $n \geq 2$ , and let $b_{n} = e_{n} + 2$ . Then $\sum_{k \leq n} e_{k} = (n + 1)!$ , so (4) holds, and $e_{n} < b_{n}$ . Nevertheless

\sum_{n \geq 1} \frac{a_{n} + e_{n}}{Q_{n}} = \sum_{n \geq 1} \frac{b_{n} - 1}{Q_{n}} = 1.

Here $gcd (b_{n}, e_{n}) \leq 2$ . This example separates rapid denominator growth from the arithmetic obstruction used in Theorem 3.6.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

Prefer the manuscript?

PDFOpen the PDF, 38 pages TeXOpen the LaTeX source

Equations typeset from the exact TeX. Rendered from the LaTeX at sha256:83e0788b73c2144a. It matched the published source manifest, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.