Determinant of matrix with entries indicating primality of i^2 + j^2

Conjecture: a(n) = 0 for no n > 28. - _Zhi-Wei Sun_, Aug 26 2013

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-71524,
  title        = {Determinant of matrix with entries indicating primality of i^2 + j^2},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-71524}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Conjecture: $a (n) = 0$ for no $n > 28$ .

_Zhi-Wei Sun_, Aug 26 2013

conjecture2. Conjecture (Generalization): For every $m \in N$ , the determinant generalDet m n is nonzero for all sufficiently large $n$ .

_Zhi-Wei Sun_, Aug 26-27 2013

The sequence $a (n)$ is the determinant of the $n \times n$ matrix $M$ defined by $M (i, j) = 1$ if $i^{2} + j^{2}$ is prime, and $0$ otherwise, where $1 \leq i, j \leq n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«71524» (2 statements).

theorem conjecture1 (n : ℕ) (hn : 28 < n) : a n ≠ 0

theorem conjecture2 (m : ℕ) : ∃ N : ℕ, ∀ n : ℕ, N < n → generalDet m n ≠ 0

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A071524

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.