Pith Number

pith:HYOWWEPY

pith:2026:HYOWWEPYKZGFN4H5PS7YPKC233

not attested not anchored not stored refs pending

On the average number of representations of an integer as a sum of polynomials computed at prime values

Alessandra Migliaccio, Alessandro Zaccagnini

The average number of representations of an integer as a sum of j values of a degree-k polynomial at prime powers follows a positive asymptotic for j at least k.

arxiv:2601.22822 v2 · 2026-01-30 · math.NT

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\usepackage{pith}
\pithnumber{HYOWWEPYKZGFN4H5PS7YPKC233}

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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 study the average number of representations of an integer n as n = φ(n1) + … + φ(nj), for polynomials φ ∈ ℤ[n] with ∂φ = k ≥ 1, lead(φ) = 1, j ≥ k, where ni is a prime power for each i.

C2weakest assumption

That the analytic machinery (likely circle method or Vinogradov-type estimates) from the cited prior works extends uniformly to arbitrary k and all j ≥ k without additional restrictions on the polynomial or error terms.

C3one line summary

Asymptotic formulas are established for the average number of ways to write n as sum of j monic polynomial values at prime powers, generalizing earlier cases for cubics and monomials.

Receipt and verification

First computed	2026-05-18T02:45:05.667209Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3e1d6b11f8564c56f0fd7cbf87a85adee84639cb78e44078aa6739203f9877fc

Aliases

arxiv: 2601.22822 · arxiv_version: 2601.22822v2 · doi: 10.48550/arxiv.2601.22822 · pith_short_12: HYOWWEPYKZGF · pith_short_16: HYOWWEPYKZGFN4H5 · pith_short_8: HYOWWEPY

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/HYOWWEPYKZGFN4H5PS7YPKC233 \
  | 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: 3e1d6b11f8564c56f0fd7cbf87a85adee84639cb78e44078aa6739203f9877fc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "40ac21c7e8d49a777b4010abef68405c930e232ab5b6118afb5339426a53260b",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-01-30T10:47:09Z",
    "title_canon_sha256": "bb0be3600fa6e60819f4d3012112a1ea30be0f10b810b9845effe71ba5f1c17a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.22822",
    "kind": "arxiv",
    "version": 2
  }
}