Let $A$ be an infinite sequence of integers such that every $n \in A + A$ is squarefree. How fast must $A$ grow?

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number theory · open · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write the sequence as (A={a_1<a_2<\cdots}\subset \mathbb Z). Your condition is that **every pairwise sum** (a_i+a_j) [[nomath]](including $2a_i$)[[/nomath]] is squarefree.

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llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

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A392164 — a(n) is the size of the largest subset S of {1,...,N} such that every element of S+S is squarefree.1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,

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#1109Let

f (N)

be the size of the largest subset

A \subseteq {1, \dots, N}

such that every

n \in A + A

is squarefree. Estimate

f (N)

. In particular, is it true that

f (N) \leq N^{o (1)}

, or even

f (N) \leq (lo g N)^{O (1)}

?A392164

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