The circulant Hadamard conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The circulant Hadamard conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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The circulant Hadamard and Barker-sequence conjectures. Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $n\in\{2,3,4,5,7,11,13\}$.

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LEVEL 1 The circulant Hadamard conjecture

released 2026-09-23 | 1 theorem · 4 lemmas · 8 proofs · 6,391 words | PLAY LEVEL 1 » (pdf)

We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.

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