Pith Number

pith:2AAHHZ2E

pith:2026:2AAHHZ2E4A4MKFRPTRP7RG6PN5

not attested not anchored not stored refs resolved

Counting solutions to the quadratic determinant equation

Akshat Mudgal, Jonathan Chapman

When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms.

arxiv:2605.15434 v1 · 2026-05-14 · math.NT

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\usepackage{pith}
\pithnumber{2AAHHZ2E4A4MKFRPTRP7RG6PN5}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

When h = N^2 + O(N), the number of solutions admits an asymptotic formula with square-root cancellation error terms obtained by exploiting symmetry via Ramanujan sums and bypassing Kloosterman sum bounds.

C2weakest assumption

The additional symmetry present precisely when h = N^2 + O(N) permits direct use of Ramanujan sums to achieve square-root cancellation without relying on general Kloosterman bounds.

C3one line summary

Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.

References

18 extracted · 18 resolved · 0 Pith anchors

[1] M. Afifurrahman,A uniform formula on the number of integer matrices with given determinant and height, J. Number Theory281(2026), 741–770 2026

[2] Apostol,Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer- Verlag, New York-Heidelberg, 1976 1976

[3] J. Chapman, A. Mudgal,On commuting integer matrices, arXiv:2504.15839, to appear in Trans. Amer. Math. Soc

[4] J. Chapman, A. Mudgal,Counting2×2integer matrices with a given determinant, arXiv:2509.20259

[5] J.-M. Deshouillers, H. Iwaniec,An additive divisor problem, J. London Math. Soc. (2)26(1982), no. 1, 1–14 1982

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:00:58.440008Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d00073e744e038c5162f9c5ff89bcf6f56d591d27c31035cf61acd7ef83a2b2d

Aliases

arxiv: 2605.15434 · arxiv_version: 2605.15434v1 · doi: 10.48550/arxiv.2605.15434 · pith_short_12: 2AAHHZ2E4A4M · pith_short_16: 2AAHHZ2E4A4MKFRP · pith_short_8: 2AAHHZ2E

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/2AAHHZ2E4A4MKFRPTRP7RG6PN5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d00073e744e038c5162f9c5ff89bcf6f56d591d27c31035cf61acd7ef83a2b2d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "923c86fc0e7aea428b4c9c9477cbaeea5b08e23bd608e2c6ecfeeecc3081db03",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-14T21:26:51Z",
    "title_canon_sha256": "7977e62c51996c1caf8eec07fbc57eccfc08fb03696928e4680bc6257c4017da"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15434",
    "kind": "arxiv",
    "version": 1
  }
}