E368: “Are infinitely many consecutive products unusually smooth?”

Open

Statement	User	Model	Harness	Time
2)V2The largest prime factor of 48·49 is smaller than (log 48)^3.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every ε > 0, are there infinitely many n for which the largest prime factor of n(n+1) is smaller than (lo…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The largest prime factor of 48·49 is smaller than (log 48)^3.

kernel-checked, filed Tue Aug 25 2026 06:03:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete n=48, ε=1 instance of the root's exact largest-prime-factor and real-power predicate.

1) V1 For every ε > 0, are there infinitely many n for which the largest prime factor of n(n+1) is smaller than (log n)^(2+ε)?

open, filed Tue Aug 25 2026 06:00:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The formal statement is the explicit conjectural component of the broader 'how large' question. A concrete epsilon=1 member n=48 kernel-checks, including maxPrimeFac(48*49)=7 and 7<(log 48)^3; an independent encoding is definitionally equal; nine content-free bridges are rejected. Full routes checked smooth-pair distribution, factorial/primorial/CRT shifts, Pell and S-unit families, ABC/Pasten methods, and definition degeneracies.

Scope. Erdős's explicit infinitely-many upper-example conjecture from problem 368; natural n≥2, genuine maximum prime factor from the factor list, natural logarithm, and real exponentiation.

E368:Erdős Problem #368 “Are infinitely many consecutive products unusually smooth?”

E368: “Are infinitely many consecutive products unusually smooth?”