Pith Number

pith:X3B6JQOU

pith:2026:X3B6JQOU6U635QVPKRQUF4GAMV

not attested not anchored not stored refs pending

Multiple Gauss sums

Jianya Liu, Sizhe Xie

A bound on multiple Gauss sums proves that nonsingular form systems have prime solutions when the number of variables meets or exceeds D squared times 4 to the D plus 2 times R to the fifth.

arxiv:2604.03347 v2 · 2026-04-03 · math.NT

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\usepackage{pith}
\pithnumber{X3B6JQOU6U635QVPKRQUF4GAMV}

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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

✓ Portable graph bundle live · download bundle · merged state

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

We prove that the system F(x)=0 is solvable in primes provided that s ≥ D² 4^{D+2} R^5, where F consists of R nonsingular forms of differing degrees with maximum degree D.

C2weakest assumption

The system of forms is nonsingular; the proof relies on this algebraic condition to control the singular series or major arcs in the analytic argument.

C3one line summary

New bound on multiple Gauss sums improves the Birch-Goldbach result: nonsingular systems of R forms of max degree D in s variables have prime solutions when s ≥ D² 4^{D+2} R^5.

Receipt and verification

First computed	2026-05-20T00:03:10.454398Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bec3e4c1d4f53dbec2af546142f0c06551e6b6f6868e0ebb035e4c4442464e61

Aliases

arxiv: 2604.03347 · arxiv_version: 2604.03347v2 · doi: 10.48550/arxiv.2604.03347 · pith_short_12: X3B6JQOU6U63 · pith_short_16: X3B6JQOU6U635QVP · pith_short_8: X3B6JQOU

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/X3B6JQOU6U635QVPKRQUF4GAMV \
  | 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: bec3e4c1d4f53dbec2af546142f0c06551e6b6f6868e0ebb035e4c4442464e61

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0aa73b3a500a57e6f4ff7ea7b4872172029ab445ce414544126752e5521dac7b",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-03T10:38:32Z",
    "title_canon_sha256": "4e628f608e030f9b97f28822412875183c276145b12904b389819edc7b750c47"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.03347",
    "kind": "arxiv",
    "version": 2
  }
}