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

Limits on methods

Status is as stated in each note: L for checked in Lean there, O for an ordinary proof there, C for cited there.

Note	Location	Statement
#68 [29]	Section 6	Under the displayed cancellation hypotheses, the integer-gap comparison fails at cutoffs $N = D + O (1)$ as the cancellation cutoff $D \to \infty$ . Small tails alone do not give the strict comparison.	O
#243 [30]	Proposition 17	A counterexample has errors that are eventually nonzero, relatively small, with unbounded negative parts. This necessary profile is formalised; the comparison with scalar profiles and the need for denominator compatibility are ordinary discussion.	L, O
#249 [31]	Section 5	A rational series with the totient’s values at odd indices, within $2$ at even indices, and sum $5 / 4$ . Positive tail differences need not be nonintegral.	L
#249 [31]	Theorem 9	Every admissible rank-one quotient stays more than $21 / 320$ from its target.	L
#251 [32]	Proposition 1, Corollary 2	For every allowance $f (n) \to \infty$ , sparse nonnegative corrections reach every real in an interval while every fixed modulus eventually divides both the corrections and their cumulative sums. A bounded allowance is impossible under these congruences. Sources: [11], [12], [13], [38].	O
#257 [33]	Section 6	The small-displacement quantity stays above $1 / 2$ at full support, where the value is irrational [37]. No proof covering full support can rest on it.	L, O
#257 [33]	Section 3	Every positive divisor cover costs at least $e$ times the mean of $\log^{+}$ of its multiplicity. The averaging method cannot reach the prime support, where irrationality is known at base $2$ [15].	O
#269 [34]	Theorem 1	Nonsingular minors of every order: no finite sum of products separates one exponent from the other two. Fan posted the two-prime separation [16]; the three-prime statement is the note’s.	L
#1049 [35]	Theorem 5	One family of nonzero integer-polynomial linear forms with common leading degree, coefficient-height and decay bounds at every fixed real $x > 1$ has $σ \leq δ$ ; the sufficient cutoff $σ / (σ + δ)$ supplied by those estimates is at most $1 / 2$ . Ordinary proof, with the contradiction step and the comparison with $1 / 2$ checked in Lean.	O, L
#1049 [35]	Theorem 7	The stated clearing conditions cannot be met at base $3 / 2$ .	L

Theorem 6.1(ii) explains interval filling in the #251 construction. The #249 countermodel is a separate explicit construction with its own preserved identities. The #1049 restriction does not exclude stronger base-specific estimates, different families at different bases, or forms involving several target values. The others bound a method. We tried to state one inequality that covers #1049 Theorem 5 and the cover cost of #257, a cost of clearing denominators against the decay gained, and did not find a formulation that survives both sets of hypotheses. We do not claim the rows share a cause.

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.

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