Discoveries / number-theory

Local limits along squares and prime values of digital functions

New theorems, not a formalisation of known results. In base ten the bare existence of a square with any admissible digit sum was recorded in the recreational literature (Murthy–Ashbacher 2005); the theorem here is the quantitative all-base version — roots coprime to b (excluding their trailing-zero trick), exp(c_b√q) representations of controlled size — and that statement is what is machine-checked, with effective but non-optimal constants. The same paper proves a local limit theorem for g(n²) for arbitrary digit weights in every base including binary, and sieves the values: level of distribution 1/2, P₃ and P₂ values, an unconditional Mertens-type prime-value law, a joint Erdős–Kac theorem. Caveats: the two shrinking-frequency estimates behind the local theorem enter the Lean development only as transcribed definitions; the sieve and asymptotic results are unformalised; Corollary 1.9 is formalised in bases 2 and 3 only; the analogous theory along squares of primes remains open.