Helleseth's three-valued conjecture
Let be a prime, with , , and the canonical additive character of . For a positive integer with , put . Helleseth conjectured that the spectrum never has exactly three distinct values; equivalently, the cross-correlation of a -ary m-sequence of period with its -decimation is never three-valued. Is that true?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Finite fields; Weil sums; sequence cross-correlation and coding theory
- Posed by
- Tor Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math. 16 (1976), 209-232, Conjecture 5.2
- Year posed
- 1976
- Years open
- 50y
- Solved
- 2026-10-23
- Model
- GPT-6 Astra; Claude Fable 5.1
- Vendor
- OpenAI; Anthropic
- Collaborators
- Mingchang Liu
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1 rules out an exactly three-valued power-permutation Weil spectrum in every characteristic when the extension degree is a power of two, completing the previously open cases in characteristic at least five. Theorem 1.2 proves that, in any degree, both nonzero values of a three-valued spectrum over satisfy . The new elementary step excludes a common nonzero residue modulo for all nonempty fibers of in odd characteristic. Corollary 3.2 removes a symmetry hypothesis from a quadratic-subfield obstruction. The work does not classify three-valued spectra in general degrees, and the proofs do not rely on finite enumeration.
What the AI did
The manuscript reports collaboration between Mingchang Liu and GPT 6 Astra and Claude Fable 5.1 in mathematical development, argument checking, and writing refinement. The author takes responsibility for the results and manuscript. Separate AI review reports assessed the argument and literature; these are not human expert certification.
Verification
No independent mathematician has checked this yet. Read by this site on 3 November 2026. The statement matches Helleseth's Three-Valued Conjecture as stated in Katz's survey (arXiv 1805.10452) and Open Problem 1 of Helleseth and Li (arXiv 2407.16072); the 1976 paper itself was not opened. The new step, Lemma 2.1 (no common nonzero residue mod p for the nonempty fibres of x^d+(1-x)^d, d odd), was read line by line and found correct; the identity K(r) = N_b(r) behind it and the lemma itself were checked by computer for every odd d < 2p and every prime p from 5 to 43, with no counterexample. The reductions to Katz and Katz-Langevin were not re-derived. A separate computation written here of the full Weil spectrum for every invertible exponent over F_{5^4}, F_{7^4} and F_{3^8} found no three-valued spectrum, while controls over F_{5^3} and F_{3^3} found the known three-valued exponents. These are finite checks. No Lean formalisation.
Sources
Submitted by SilentIbis759 on