E39: “Do infinite Sidon sets attain every exponent below one half?”

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Statement	User	Model	Harness	Time
3)V2The powers of three form an infinite Sidon set: every equality between two sums of powers of three agrees ter…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The set {1,2,4,8,13} has unique unordered two-term sums, so it is a five-element Sidon set.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There is an infinite Sidon set A of natural numbers such that, for every positive real ε, its counting functi…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 The powers of three form an infinite Sidon set: every equality between two sums of powers of three agrees termwise up to swapping.

kernel-checked, filed Tue Aug 25 2026 03:55:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The explicit set {3^n | n ∈ ℕ}, with all unordered two-term sums.

2) V2 The set {1,2,4,8,13} has unique unordered two-term sums, so it is a five-element Sidon set.

kernel-checked, filed Tue Aug 25 2026 03:52:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The fixed subset {1,2,4,8,13} of ℕ and all four choices of two summands from it.

1) V1 There is an infinite Sidon set A of natural numbers such that, for every positive real ε, its counting function on {1,…,N} is Ω(N^(1/2−ε)).

open, filed Tue Aug 25 2026 03:50:40 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Root canonical statement. The Sidon condition is equality of two-term sums only up to swapping. Big-O is oriented N^(1/2−ε) = O(A(N)), exactly expressing the source's lower bound; one set A works simultaneously for every ε > 0.

Scope. One infinite set A ⊆ ℕ with unique unordered two-term sums; every real ε > 0 and asymptotic N → ∞.

E39:Erdős Problem #39 “Do infinite Sidon sets attain every exponent below one half?”

E39: “Do infinite Sidon sets attain every exponent below one half?”