Pith Number

pith:QPF74FBE

pith:2026:QPF74FBEHHE3GQYIMDBB3WSUR3

not attested not anchored not stored refs pending

Explicit Prime Densities for the Rank of Appearance in Lucas Sequences

Joaquim Cera Da Concei\c{c}\~ao

Closed-form formulas exist for the Dirichlet density of primes p where a fixed d divides the rank of appearance in any Lucas sequence U.

arxiv:2604.20014 v2 · 2026-04-21 · math.NT

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\pithnumber{QPF74FBEHHE3GQYIMDBB3WSUR3}

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Record completeness

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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

We derive closed-form formulas for the Dirichlet density of primes p for which d∣ρ_U(p), where d≥1 is a fixed integer. Our results complete the work of Sanna (2022) by covering all U and all d≥1.

C2weakest assumption

The assumption that uniform closed-form expressions exist and can be derived for every Lucas sequence U (including degenerate cases) and every d, relying on the standard algebraic properties of the discriminant and the recurrence without exceptional cases that break the formulas.

C3one line summary

Closed-form Dirichlet density formulas are derived for primes p where d divides ρ_U(p) in Lucas sequences U, covering all U and all d ≥ 1.

Receipt and verification

First computed	2026-05-22T01:03:19.463250Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

83cbfe142439c9b3430860c21dda548ed894070da6a8c73045391d7f7af84806

Aliases

arxiv: 2604.20014 · arxiv_version: 2604.20014v2 · doi: 10.48550/arxiv.2604.20014 · pith_short_12: QPF74FBEHHE3 · pith_short_16: QPF74FBEHHE3GQYI · pith_short_8: QPF74FBE

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/QPF74FBEHHE3GQYIMDBB3WSUR3 \
  | 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: 83cbfe142439c9b3430860c21dda548ed894070da6a8c73045391d7f7af84806

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "efb246e40f3b25036a42ed0c80a34468ae6749dd5dcfb4e5ec20824343592698",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-21T21:48:43Z",
    "title_canon_sha256": "849368343cdb48c84f19825f6594926971785ec8dd275d869deec83195241668"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.20014",
    "kind": "arxiv",
    "version": 2
  }
}