Erdős Problem 684: f(n)/log n is unbounded — a Lean 4 formalization of arXiv:2604.23784
A Lean 4 formalization, against Mathlib, of the main theorem of J. H. Bae, "Unbounded logarithmic limsup in Erdős Problem 684 via shifted carry scheduling" (arXiv:2604.23784, v3). For 0 ≤ k ≤ n write C(n,k) = u·v with u supported on the primes ≤ k and v on the primes in (k,n], and let f(n) be the least k with u > n²; Erdős asked for bounds on f(n) (Problem 684 in Bloom's database). The compared theorems state that for every ε > 0 there are infinitely many n with f(n) > (1/2 − ε)·log n·log log n / log log log n, equivalently limsup f(n)·log log log n /(log n·log log n) ≥ 1/2 (stated in the…
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