What does this research mean for the field?

Sun's 1-3-4 conjecture is true, establishing that the nonnegative integers with no representation n=x^2+y^2+z^2+w^2 (with nonnegative integral coordinates and x+3y+4z a square) are exactly 16^k*e, where k>=0 and e is in {8, 24, 40, 344}. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To prove Sun's conjecture (OEIS A335624) characterizing every nonnegative integer that cannot be represented as the sum of four squares with a constrained square combination.

September 14, 2026Open Access

Sun's 1–3–4 Conjecture: A Computer-Assisted Proof for OEIS A335624

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JPJulio Leandro Sotomayor Pimentel

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Implication

Computational proof resolves Sun's 1–3–4 conjecture on four-square representations in nonnegative integers, confirming exactly four base exceptions.

Key Points

To prove Sun's conjecture (OEIS A335624) characterizing every nonnegative integer that cannot be represented as the sum of four squares with a constrained square combination.
Applied a quaternionic framework with Sun's sign criterion, combining a norm-26 transfer controlled modulo 3 with an even-norm positivity refinement to establish analytical existence for n ≥ 17,000,000,000 (with 16 ∤ n).
Verified the remaining finite range below 1.7 × 10¹⁰ using three distinct algorithms: two bitmap sieves and an exhaustive scalar search.
Proved that the nonnegative integers lacking a representation n = x² + y² + z² + w² with x + 3y + 4z being a square are precisely 16ᵏe, where k ≥ 0 and e ∈ {8, 24, 40, 344}.
Confirmed through all three computational searches that exactly four base exceptions exist below the analytical threshold.

Cite This Study

Julio Leandro Sotomayor Pimentel (2026) studied this question.

synapsesocial.com/papers/6aa7b3aa0926e14a848b2c23 https://doi.org/10.5281/zenodo.22718411

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Implication

Computational proof resolves Sun's 1–3–4 conjecture on four-square representations in nonnegative integers, confirming exactly four base exceptions.

Sun's 1–3–4 Conjecture: A Computer-Assisted Proof for OEIS A335624

Discussion

Member takes

Implication

Key Points

Cite This Study

Sun's 1–3–4 Conjecture: A Computer-Assisted Proof for OEIS A335624

Discussion

Member takes

Implication

Key Points

Cite This Study

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