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

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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.

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In this paper

A digit that also corrects the cumulative residue

The following construction supplies the positive direction. It is useful beyond factorial weights.

Lemma 3.4. Let $w_{j} > 0$ , let positive integers $M_{j}$ satisfy $M_{j - 1} ∣ M_{j}$ , and let $F_{j} \geq 0$ . Suppose $M_{j} w_{j} \to 0$ and, for $j \geq 1$ ,

\begin{matrix} (7) & 2 M_{j} \leq M_{j - 1} \frac{w_{j - 1}}{w_{j}}, 2 M_{j - 1} \frac{w_{j - 1}}{w_{j}} \leq F_{j} . \end{matrix}

Every $y \in [M_{0} w_{0}, 2 M_{0} w_{0}]$ has a representation $y = \sum_{j \geq 1} b_{j} w_{j}$ with integers $0 \leq b_{j} \leq F_{j}$ such that

M_{j} ∣ \sum_{i \leq j} b_{i}, M_{j - 1} ∣ b_{j} .

Proof. Start with residual $R_{0} = y$ and cumulative sum $C_{0} = 0$ . Suppose $M_{j - 1} w_{j - 1} \leq R_{j - 1} \leq 2 M_{j - 1} w_{j - 1}$ and $M_{j - 1} ∣ C_{j - 1}$ . Put $u = R_{j - 1} / w_{j}$ and let $v$ be the least nonnegative residue of $- C_{j - 1}$ modulo $M_{j}$ . Choose

\begin{matrix} (8) & b_{j} = v + M_{j} ⌊ \frac{u - M_{j} - v}{M_{j}} ⌋ . \end{matrix}

Since $u \geq 2 M_{j}$ and $0 \leq v < M_{j}$ , this integer is nonnegative. The floor identity gives $M_{j} \leq u - b_{j} < 2 M_{j}$ ; also $b_{j} \leq u \leq F_{j}$ . Thus $R_{j} = R_{j - 1} - b_{j} w_{j}$ lies in $[M_{j} w_{j}, 2 M_{j} w_{j})$ , while $C_{j} = C_{j - 1} + b_{j}$ is divisible by $M_{j}$ . The incoming modulus divides both $C_{j - 1}$ and $M_{j}$ , hence also $b_{j}$ . Finally $R_{j} \to 0$ , so the partial sums converge to $y$ . ◻

The residue in (8) may depend on $y$ . Only the schedule of moduli needs to be common to all targets. Requiring target-independent ordinary block sums would impose an unnecessary restriction on the construction.

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.

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