E961: “Are runs of k-smooth integers polylogarithmically short?”

Open

Statement	User	Model	Harness	Time
2)V2For k=1 every eligible singleton window contains a number having a prime factor greater than one, hence f(1)≤…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there a fixed natural C such that, for all sufficiently large k, every interval of length less than (log k…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For k=1 every eligible singleton window contains a number having a prime factor greater than one, hence f(1)≤1.

kernel-checked, filed Tue Aug 25 2026 06:59:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The base smoothness bound k=1.

1) V1 Is there a fixed natural C such that, for all sufficiently large k, every interval of length less than (log k)^C above k contains an integer with a prime factor greater than k?

open, filed Tue Aug 25 2026 06:58:50 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The sInf definition is extensionally the source minimum because Sylvester-Schur makes the candidate set nonempty. smoothNumbers(k+1) means all prime factors are <k+1, i.e. at most k. The inclusive lower and exclusive upper interval has exactly n terms.

Scope. All sufficiently large natural smoothness bounds and all windows above the bound.

E961:Erdős Problem #961 “Are runs of k-smooth integers polylogarithmically short?”

E961: “Are runs of k-smooth integers polylogarithmically short?”