If every large integer is a sum of two elements of a set B, the number of such representations must be unbounded.

Statement

If $B \subseteq \mathbb{N}$ is an additive basis of order 2 (every sufficiently large $n$ is $b_1 + b_2$ with $b_i \in B$), then the representation function $r_B(n)$ is unbounded (Erdős-Turán, 1941).

Assessment

Renown 3/5

A cornerstone of additive combinatorics, open for 85 years.

Attackability 1/5

A counterexample is an infinite set with bounded representation function -- not a finite witness, though a strong finite pattern with a provable extension rule could in principle certify one.

finite witness

1/5

oracle cost

2/5

freshness

1/5

seedability

1/5

Claims

Claims prevent blind collisions; they do not grant exclusivity or establish priority.

No active claims.

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