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Local limits along squares and prime values of digital functions

For every base b≥2 and sufficiently large digit sum q congruent to a square modulo b−1 (automatic for b=2), the authors construct at least exp(c_b√q) squares n² of bounded size with gcd(n,b)=1 and base-b digit sum q. The explicit Lean statement gives q≥2304b⁴, n²≤b^(3q), and at least 2^(⌊√q/(36b)⌋) examples for b≥3, with a separate binary case.

Sources for this record

What was already known

A base-ten existence argument using trailing-zero padding was known. It did not require roots coprime to the base or give this quantitative all-base count.

AI involvement AI-centered

The author-submitted account credits models with jointly developing the carry-free Sidon-set construction under human direction. The author reports checking the statements and proofs; the manuscript also credits models with much of its writing.

The retained carry-free Sidon construction for the assessed quantitative digit-sum theorem was jointly worked out with models under human direction.

Importance Notable

The carry-free construction strengthens existence to an explicit all-base count with coprime roots.

Limitations

The formal check covers the prescribed-digit-sum construction, not the manuscript’s broader local-limit, sieve or prime-value theorems. The detailed model-origin account comes from the author-submitted registry entry.

Technical verification Formal (Lean 4 / mathlib kernel + human)

The author reports checking all proofs, and the specific Theorem 1.8 digit-sum construction is represented by sorry-free Lean statements with no paper-specific axioms; a registry audit compared the statement to the submitted claim. This does not elevate the broader square local-limit theorem, whose shrinking-frequency estimates remain outside the Lean proof.

Independent checking

Author review of statements and proofs as stated in the author-submitted registry account

Checker
vibefrtz, pseudonymous paper author
Scope
Theorem 1.8 quantitative all-base prescribed square-digit-sum result
Basis
Reported by the linked checker
Evidence source

Formal checking

Sorry-free formal proof with axiom audit containing only Lean's standard axioms

Checker
Lean 4/mathlib kernel on DSS.sq_digit_sum_count and sq_digit_sum_count_two
Scope
The explicit prescribed-digit-sum count with coprime roots in every base
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The author jointly worked out the Sidon construction and verified the argument, documenting understanding of the carry-free counting mechanism.

Correction and publication history