Pith Number

pith:GMUCINFV

pith:2026:GMUCINFVQOZBPZKS5LXUA4ZW56

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A density version of quaternary Goldbach problem

Meng Gao, Xiaoyang Hu

If four subsets of primes satisfy underline delta(P1) + underline delta(P2) > 1 and underline delta(P3) + underline delta(P4) > 1, then every sufficiently large even n is the sum of one prime from each subset, and the bound is sharp.

arxiv:2605.14369 v1 · 2026-05-14 · math.NT

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\usepackage{pith}
\pithnumber{GMUCINFVQOZBPZKS5LXUA4ZW56}

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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

Suppose that P1, P2, P3, P4 are four subsets of primes with underline delta(P1)+underline delta(P2)>1 and underline delta(P3)+underline delta(P4)>1. Then for every sufficiently large even integer n, there exist primes pi in Pi (i=1,2,3,4) such that n=p1+p2+p3+p4. The condition is the best possible.

C2weakest assumption

The proof assumes that the density conditions are sufficient to apply some form of the circle method or sieve to control the singular series and minor arcs for the quaternary sum; if the analytic estimates fail at the stated density threshold, the representation claim would not hold.

C3one line summary

References

19 extracted · 19 resolved · 1 Pith anchors

[1] A. Alsetri, X. C. Shao,Density versions of the binary Goldbach problem, Acta Arith.,218(2025), no. 3, 285–295 2025

[2] Gao,A density version of Waring-Goldbach problem, Int 2025

[3] Gao,On the representation of large even integers as the sum of eight primes from positive density sets, Bull 2025

[4] Green,Roth’s theorem in the primes, Ann 2005

[5] B. Green, T. Tao,The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2),167 (2008), no. 2, 481–547 2008

Receipt and verification

First computed	2026-05-17T23:39:07.849654Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

33282434b583b217e552eaef407336efa86cf418e386a9293bc067515166a474

Aliases

arxiv: 2605.14369 · arxiv_version: 2605.14369v1 · doi: 10.48550/arxiv.2605.14369 · pith_short_12: GMUCINFVQOZB · pith_short_16: GMUCINFVQOZBPZKS · pith_short_8: GMUCINFV

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/GMUCINFVQOZBPZKS5LXUA4ZW56 \
  | 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: 33282434b583b217e552eaef407336efa86cf418e386a9293bc067515166a474

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b1a100ce8c5479a902956b937016e65fc8a6ffeb5eaf8129fb597f24184117d4",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-14T04:48:38Z",
    "title_canon_sha256": "c6b59c3426ebb72cc4e677f81125f7632a045c90f15f85642e70a14b2bc1ab93"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14369",
    "kind": "arxiv",
    "version": 1
  }
}