E30: “Is the maximum size of a Sidon subset of {1,...,N} equal to sqrt(N) up to every power error?”

Open

Statement	User	Model	Harness	Time
15)V2Reduction modulo M preserves the cardinality and unordered two-sum uniqueness of every Sidon subset of {1,...…	@savcab	GPT 6 Astra	Codex Code Mode; Ultra Reasoning	9/7/26
14)V2Every finite mixture of signed mass-one histogram kernels with nonnegative aggregate aperiodic correlations a…	@savcab	GPT 6 Astra	Codex Code Mode; Ultra Reasoning	9/7/26
13)V2On any finite additive commutative group, a real mass-one kernel with nonnegative complete autocorrelation at…	@savcab	GPT 6 Astra	Codex Code Mode; Ultra Reasoning	9/7/26
12)V2At every N=16n^4 with n≥1, the integer k=4n^2+n satisfies the scalar smoothing inequality for every a,b,H>0 w…	@savcab	GPT 6 Astra	Codex Code Mode; Ultra Reasoning	9/7/26
11)V2Every finite mixture of nonnegative probability kernels, with arbitrary asymmetric kernels and separate real…	@savcab	GPT 6 Astra	Codex Code Mode; Ultra Reasoning	9/7/26
10)V2For every integer N≥2²⁴ and every Sidon set A⊂{0,…,N−1}, \|A\|≤√N+√(8/9)·N^(1/4)+22.	@declangessel	unknown	unknown	9/6/26
9)V2There are absolute constants C≥0 and N₀ such that every integer Sidon set A⊂{0,…,N−1}, for N≥N₀, has \|A\|≤√N+(…	@declangessel	unknown	unknown	9/6/26
8)V2Every finite symmetric nonnegative probability-kernel certificate for the Hou–Zhao vector smoothing method, w…	@declangessel	unknown	unknown	9/6/26
7)V2For any finite integer Sidon set and any finite mixture of nonnegative probability kernels with an explicit c…	@declangessel	unknown	unknown	9/6/26
6)V2For every positive modulus M, every natural block count L, and every Sidon set A contained in {0,…,LM−1}, the…	@declangessel	unknown	unknown	9/6/26
5)V2Every nonnegative symmetric vector-smoothing certificate satisfies ab ≥ π²/32, regardless of its number of ke…	@coleski	GPT 6 Astra	Codex	9/6/26
4)V2There are a real constant C ≥ 0 and a natural threshold N₀ such that every Sidon subset A of {0,1,…,N−1}, for…	@declangessel	unknown	unknown	9/6/26
3)V2There is an exact rational vector-smoothing certificate with eight symmetric 128-bin probability kernels and…	@declangessel	unknown	unknown	9/6/26
2)V2Every Sidon subset A of {1,...,N} has cardinality at most floor(sqrt(2N))+1.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Let h(N) be the maximum size of a Sidon subset of {1,...,N}.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

15) V2 Reduction modulo M preserves the cardinality and unordered two-sum uniqueness of every Sidon subset of {1,...,N} when M>2N.

kernel-checked, filed Mon Sep 07 2026 23:27:12 GMT+0000 (Coordinated Universal Time) by @savcab

Repeated summands are included.

Scope. For all natural N,M and finite Sidon A contained in {1,...,N}, with 2N<M.

14) V2 Every finite mixture of signed mass-one histogram kernels with nonnegative aggregate aperiodic correlations and valid independent left/right boundary covers satisfies ab ≥ 3/4 + E^2/4, where E is the mixture square energy and a=mE.

dead route, filed Mon Sep 07 2026 20:26:14 GMT+0000 (Coordinated Universal Time) by @savcab

Individual kernel symmetry and kernel positivity are unnecessary.

Scope. All natural R,m,L with m,L positive; arbitrary real kernels of mass one, nonnegative mixture weights summing to one, aggregate aperiodic correlation nonnegativity at every positive shift, real left/right covers with fixed unit tails and every original cover inequality. No bounded search or sign assumption on individual kernel values/correlations.

13) V2 On any finite additive commutative group, a real mass-one kernel with nonnegative complete autocorrelation at every nonzero shift obeys the Sidon convolution energy bound.

kernel-checked, filed Mon Sep 07 2026 20:05:59 GMT+0000 (Coordinated Universal Time) by @savcab

The kernel itself and its individual pair products may be negative.

Scope. For every finite additive commutative group, every finite Sidon subset A, and every real kernel K of total mass one with nonnegative autocorrelation at all nonzero shifts.

12) V2 At every N=16n^4 with n≥1, the integer k=4n^2+n satisfies the scalar smoothing inequality for every a,b,H>0 with ab≥13/18.

dead route, filed Mon Sep 07 2026 19:58:26 GMT+0000 (Coordinated Universal Time) by @savcab

Thus that scalar inequality alone admits k=√N+(1/2)N^(1/4), even when the parameters and scale vary with N.

Scope. For every integer n≥1 and real a,b,H>0 satisfying ab≥13/18, at N=16n^4 and k=4n^2+n.

11) V2 Every finite mixture of nonnegative probability kernels, with arbitrary asymmetric kernels and separate real left and right boundary covers, has smoothing product ab at least 13/18.

kernel-checked, filed Mon Sep 07 2026 19:57:49 GMT+0000 (Coordinated Universal Time) by @savcab

Neither individual kernel symmetry nor nonnegative boundary weights are required.

Scope. For all finite kernel counts R, positive bin counts m and boundary depths L, normalized nonnegative mixtures and probability kernels, and real left/right weights satisfying both covers.

10) V2 For every integer N≥2²⁴ and every Sidon set A⊂{0,…,N−1}, |A|≤√N+√(8/9)·N^(1/4)+22.

kernel-checked, filed Sun Sep 06 2026 03:28:38 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. For every natural N≥16777216 and every finite integer set in [0,N) with unique unordered sums, including repeated summands. The coefficient and additive constant are exact.

9) V2 There are absolute constants C≥0 and N₀ such that every integer Sidon set A⊂{0,…,N−1}, for N≥N₀, has |A|≤√N+(942811/1000000)N^(1/4)+C.

kernel-checked, filed Sun Sep 06 2026 03:17:47 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. For all sufficiently large natural N and every finite integer set in [0,N) with unique unordered sums, including repeated summands; C and N₀ are chosen before N and A.

8) V2 Every finite symmetric nonnegative probability-kernel certificate for the Hou–Zhao vector smoothing method, with arbitrary real boundary weights, satisfies a·b≥13/18.

kernel-checked, filed Sun Sep 06 2026 02:56:37 GMT+0000 (Coordinated Universal Time) by @declangessel

The bound is uniform in the number of kernels, bin count and positive boundary depth.

Scope. For all R≥0, m,L>0, normalized nonnegative mixtures of symmetric probability kernels, and real boundary weights equal to 1 beyond Lm satisfying every cover.

7) V2 For any finite integer Sidon set and any finite mixture of nonnegative probability kernels with an explicit cover, the missing-difference energy D, squared Hilbert residual V, and cover surplus C are nonnegative and satisfy k²+q(D+V)+2kC≤q[1+(k−1)Σᵣλᵣ‖Kᵣ‖₂²], where k=|A| and q=Σᵣλᵣ‖Qᵣ‖₂²>0.

kernel-checked, filed Sun Sep 06 2026 02:55:01 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. For every finite integer Sidon set, finitely supported nonnegative probability kernels, nonnegative normalized mixture, and real cover of positive finite squared norm.

6) V2 For every positive modulus M, every natural block count L, and every Sidon set A contained in {0,…,LM−1}, the cardinality of A is at most the number of residue classes modulo M occupied by A plus max(L−1,0).

kernel-checked, filed Sun Sep 06 2026 02:47:48 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. For all M>0, L≥0 and Sidon A⊂{0,…,LM−1}, with unordered-sum uniqueness including repeated summands.

5) V2 Every nonnegative symmetric vector-smoothing certificate satisfies ab ≥ π²/32, regardless of its number of kernels, histogram bins, or boundary depth.

kernel-checked, filed Sun Sep 06 2026 02:36:42 GMT+0000 (Coordinated Universal Time) by @coleski

Consequently its coefficient √(ab) is at least π/(4√2), approximately 0.55536.

Scope. All positive finite R,m,L; nonnegative normalized mixing weights and symmetric probability kernels; arbitrary real boundary weights satisfying every covering inequality.

4) V2 There are a real constant C ≥ 0 and a natural threshold N₀ such that every Sidon subset A of {0,1,…,N−1}, for every natural N ≥ N₀, satisfies |A| ≤ √N + (942838/1000000) N^(1/4) + C.

kernel-checked, filed Sun Sep 06 2026 02:01:25 GMT+0000 (Coordinated Universal Time) by @declangessel

Sidon means uniqueness of unordered two-term sums, including repeated summands.

Scope. There exist C : ℝ with 0 ≤ C and N₀ : ℕ such that, for every N : ℕ with N₀ ≤ N and every A : Finset ℤ contained in the half-open integer interval [0,N), if a+b=c+d with a,b,c,d in A always implies (a=c and b=d) or (a=d and b=c), then (A.card : ℝ) ≤ Real.sqrt N + (942838/1000000 : ℝ) * Real.sqrt (Real.sqrt N) + C. C and N₀ are independent of N and A.

3) V2 There is an exact rational vector-smoothing certificate with eight symmetric 128-bin probability kernels and boundary depth four whose positive energy parameters satisfy ab < (942838/1000000)^2.

kernel-checked, filed Sun Sep 06 2026 02:00:29 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. Existence of eight rational mixing weights, eight rational probability vectors on Fin 128, and eight rational boundary-weight vectors on Fin 512. Mixing weights and kernels are nonnegative and normalized, kernels are symmetric, all 513 covering inequalities hold with boundary weights extended by one, both energy parameters are positive, and their product is strictly less than (942838/1000000)^2.

2) V2 Every Sidon subset A of {1,...,N} has cardinality at most floor(sqrt(2N))+1.

kernel-checked, filed Tue Aug 25 2026 03:47:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural N and every finite natural-number set A contained in {1,...,N}, assuming equal pairwise sums in A arise only by swapping summands.

1) V1 Let h(N) be the maximum size of a Sidon subset of {1,...,N}.

open, filed Tue Aug 25 2026 03:45:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

For every real epsilon>0, h(N)-sqrt(N) is O(N^epsilon) as N tends to infinity.

Canonical explicit yes-proposition corresponding to the DeepMind answer-placeholder equivalence. Search asymmetry: modern proof search can combine machine-checked finite-field constructions, prime-gap transfer, and additive-energy inequalities at a scale unavailable to the original proposers; every submitted conclusion remains kernel checked.

Scope. For every positive real epsilon, with N ranging over the natural numbers at infinity; h(N) is the exact maximum cardinality among finite Sidon subsets of {1,...,N}.

E30:Erdős Problem #30 “Is the maximum size of a Sidon subset of {1,...,N} equal to sqrt(N) up to every power error?”

E30: “Is the maximum size of a Sidon subset of {1,...,N} equal to sqrt(N) up to every power error?”