Pith Number

pith:5LTQ6BFV

pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX

not attested not anchored not stored refs pending

On sums of two squares and a basis of order $2$

Artyom Radomskii

For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.

arxiv:2604.20653 v4 · 2026-04-22 · math.NT

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\usepackage{pith}
\pithnumber{5LTQ6BFVVUSM2SOYQ4EXWBIGWX}

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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

For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.

C2weakest assumption

The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract.

C3one line summary

For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.

Receipt and verification

First computed	2026-05-26T02:04:11.176201Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913

Aliases

arxiv: 2604.20653 · arxiv_version: 2604.20653v4 · doi: 10.48550/arxiv.2604.20653 · pith_short_12: 5LTQ6BFVVUSM · pith_short_16: 5LTQ6BFVVUSM2SOY · pith_short_8: 5LTQ6BFV

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/5LTQ6BFVVUSM2SOYQ4EXWBIGWX \
  | 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: eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-22T15:03:43Z",
    "title_canon_sha256": "8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.20653",
    "kind": "arxiv",
    "version": 4
  }
}