Two smooth summands for every integer

• Slug: two-smooth-summands
• List: P12
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Additive number theory
• Sources: Green 100 #59; Erdős Problem #334
• Started: 2026-08-17

Statement

For fixed ε>0, it remains unknown whether every n≤N is a sum of two integers whose prime factors are all at most N^ε, for all sufficiently large N.

Tonight

A certified finite residue cover, an explicit ε with a checkable covering template, or a documented obstruction. Isolated small-N tables are an incomplete search unless they imply an infinite covering. Fetch Green #59 and Erdős #334 before searching.

Outcome (2026-08-17)

Incomplete search. No infinite covering and no exponent below Balog $4/(9\sqrt e)$. Certified: the trivial $2\sqrt n+1$ template, the negative-pseudosquare lemma, $F(131486759)=83$, $G(y)$ through $y=79$, and exact exception prefixes for $n^{1/2}$, $n^{2/5}$, $n^{1/3}$ that do not lift. Details in ATTACK.md / RESEARCH.md.

Outcome (2026-08-27)

Still incomplete. Closed-form templates (square, triangular, cube, largest power of two, floor-divisor) all have holes persisting through $n=20000$ at every tested exponent below $1/2$. The floor-divisor and power-of-two formulas each have an explicit infinite failure family. Polynomial values of degree $d\ge 2$ cannot beat the square covering by size of remainder. Two-factor search with $u\le n^{1/5}$ matches the known $F$ exception prefixes (sixteen at $2/5$, last $479$; seventy-six at $1/3$, last $18191$) and does not lift. No exponent below Balog. Green #59 / Erdős #334 still open. Code in compute/q1/.