Pith Number

pith:BBPCGHJI

pith:2026:BBPCGHJI72E77USBZSOC543JBS

not attested not anchored not stored refs pending

A note on Bremner's conjecture and uniformity

Hector Pasten, Natalia Garcia-Fritz

If ranks of elliptic curves over the rationals are uniformly bounded, then arithmetic progressions of their rational x-coordinates have uniformly bounded lengths.

arxiv:2604.04850 v2 · 2026-04-06 · math.NT

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\usepackage{pith}
\pithnumber{BBPCGHJI72E77USBZSOC543JBS}

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Record completeness

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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

if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions.

C2weakest assumption

The uniform Mordell--Lang conjecture for curves due to Dimitrov--Gao--Habegger, invoked to replace the earlier Nevanlinna-theoretic argument.

C3one line summary

A more direct proof establishes that uniform boundedness of ranks of rational elliptic curves implies uniform boundedness of lengths of arithmetic progressions in x-coordinates of rational points, relying only on the uniform Mordell-Lang conjecture for curves.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:05:44.368078Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

085e231d28fe89ffd241cc9c2ef3690cb081a731aa896ccb1321ae19300e09b9

Aliases

arxiv: 2604.04850 · arxiv_version: 2604.04850v2 · doi: 10.48550/arxiv.2604.04850 · pith_short_12: BBPCGHJI72E7 · pith_short_16: BBPCGHJI72E77USB · pith_short_8: BBPCGHJI

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/BBPCGHJI72E77USBZSOC543JBS \
  | 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: 085e231d28fe89ffd241cc9c2ef3690cb081a731aa896ccb1321ae19300e09b9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e2ecb6be54c138f7c2d2ddb3d6e6a0328d153c81bf80dc6fa08283d703939505",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-06T16:52:27Z",
    "title_canon_sha256": "01db2af45754eae55986ae433b2dcc0d6ac6c72e33c9fc78dac5c6f898a313b0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.04850",
    "kind": "arxiv",
    "version": 2
  }
}