Pith Number

pith:7DR4BPYX

pith:2026:7DR4BPYXL7DF2NPKRRU43PBJVC

not attested not anchored not stored refs pending

Matchable numbers

Carl Pomerance, Nathan McNew

A natural number is matchable when its divisors can be bijectively paired with 1 through tau(n) so each pair is coprime, and the paper proves that this property holds for all squarefree numbers while the full set of matchable numbers has a

arxiv:2604.05304 v2 · 2026-04-07 · math.NT · math.CO

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\usepackage{pith}
\pithnumber{7DR4BPYXL7DF2NPKRRU43PBJVC}

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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 show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable.

C2weakest assumption

The proofs rely on the existence of the required bijection for squarefree numbers and on standard techniques for establishing asymptotic densities of sets defined by divisor conditions; if the combinatorial matching for squarefree n fails in some cases or if the density calculation involves unstated exclusions, the claims would not hold.

C3one line summary

Matchable numbers, defined via a coprime bijection between divisors and {1 to tau(n)}, have a positive asymptotic density that the authors compute, and all squarefree numbers are matchable.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-26T01:02:34.052167Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f8e3c0bf175fc65d35ea8c69cdbc29a8b474deafe990e565d1ca5ae63d476d2a

Aliases

arxiv: 2604.05304 · arxiv_version: 2604.05304v2 · doi: 10.48550/arxiv.2604.05304 · pith_short_12: 7DR4BPYXL7DF · pith_short_16: 7DR4BPYXL7DF2NPK · pith_short_8: 7DR4BPYX

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/7DR4BPYXL7DF2NPKRRU43PBJVC \
  | 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: f8e3c0bf175fc65d35ea8c69cdbc29a8b474deafe990e565d1ca5ae63d476d2a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "cacd6ec2f396fa3a997735560d39653bcdb8d6c8a0257e59515860feb1b2beca",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-07T01:18:01Z",
    "title_canon_sha256": "8b5e50e95782cf1b21fa14487318c746e0a9324fefd8a3d343d138c8cd8674be"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.05304",
    "kind": "arxiv",
    "version": 2
  }
}