Pith Number

pith:JPKCDNWH

pith:2026:JPKCDNWHX2UNKVZICVHQKMMUSO

not attested not anchored not stored refs pending

Sum of consecutive powers as a perfect power

Angelos Koutsianas, Nikos Tzanakis

For k ≡ 2 mod 4 the equation x^k + (x+1)^k = y^n (n ≥ 3) has only the solutions x = 0, -1 when 6 ≤ k ≤ 100 or k has an odd prime factor ≡ 3 mod 4.

arxiv:2605.18348 v1 · 2026-05-18 · math.NT

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\usepackage{pith}
\pithnumber{JPKCDNWHX2UNKVZICVHQKMMUSO}

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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 prove that the only solutions are for x=0, -1 when 6≤k≤100 or for a k with odd prime factors congruent to 3 mod 4.

C2weakest assumption

The linear forms in logarithms and the modular method together produce effective bounds that cover all possible solutions without exception for the stated range of k (abstract, methods paragraph).

C3one line summary

For k ≡ 2 mod 4 the equation x^k + (x+1)^k = y^n (n ≥ 3) has only the solutions x = 0, -1 when 6 ≤ k ≤ 100 or k has an odd prime factor ≡ 3 mod 4.

Receipt and verification

First computed	2026-05-20T00:05:56.419654Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4bd421b6c7bea8d55728154f05319493b191a55fb7cb1cf27177945f9dc2eef4

Aliases

arxiv: 2605.18348 · arxiv_version: 2605.18348v1 · doi: 10.48550/arxiv.2605.18348 · pith_short_12: JPKCDNWHX2UN · pith_short_16: JPKCDNWHX2UNKVZI · pith_short_8: JPKCDNWH

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/JPKCDNWHX2UNKVZICVHQKMMUSO \
  | 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: 4bd421b6c7bea8d55728154f05319493b191a55fb7cb1cf27177945f9dc2eef4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "00a4524dce49c1f447559c017490e00baf1621a1eb4c0fbd62d1d9a157ef7df8",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-18T13:06:04Z",
    "title_canon_sha256": "687d199697ed8fd53bdfbcc0490fcfeb36c8b34949733639e401394309936b9d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.18348",
    "kind": "arxiv",
    "version": 1
  }
}