bibkey: oeis2025a115770
authors: Antti Karttunen
year: 2025
title: "OEIS A115770: seven and five versus binary carryless multiplication"
doi: null
url: https://oeis.org/A115770
claim: "A115770 names the natural numbers k satisfying 7*k = A048720(11,k) and conjectures that they are exactly those satisfying 5*k = A048720(13,k)."
strata_touched:
  - D5/S1/Digit/Admissibility/CarrylessSevenFiveEquivalence
license: citation-only
triage: anchor

Seven and five versus binary carryless multiplication

A115770, revision 22, NAME defines the numbers k with 7*k = A048720(11,k). The COMMENTS conjecture reads: “Conjecture: also numbers k such that 5*k = A048720(13,k). This has been checked for all k in range 0 .. 2^25.” It is attributed to Antti Karttunen, Dec 21 2025. The quantifier includes zero and has no upper bound.

A048720, revision 62, is binary polynomial multiplication over GF(2). The binary digit of degree t is the remainder modulo two of the sum of products of input digits of degrees i and t-i, for 0 ≤ i ≤ t. All other multiplications in the conjecture are ordinary natural-number multiplications. The row-five/row-seven conjecture in A115872 is the same question, not a separate assertion.

The original locators are https://oeis.org/A115770/internal and https://oeis.org/A048720/internal. The source entry records a conjecture and a finite check; the finite check is not a universal proof. No worldwide priority claim is made.

Verified locator

• URL: https://oeis.org/A115770
• Question locator: https://oeis.org/A115770/internal — NAME gives 7*k = A048720(11,k); COMMENTS states the conjecture 5*k = A048720(13,k) and the finite check through 2^25, revision 22.
• Operation locator: https://oeis.org/A048720/internal — NAME and FORMULA define binary polynomial multiplication over GF(2), revision 62.

Literature/search scope

The bounded source check used A115770 revision 22 and A048720 revision 62, a lexical repository search including private D5 sources and pinned mathlib at 56a005855ea17e0b0958f73ec801ae140b0a1679, and a caller-qualified search covering two AI-OEIS papers, 492 pinned metadata samples and their mapping, and 38 published APN files. No exact answer was found in that scope. The unpublished APN remainder, private or unindexed material, and semantically equivalent answers were excluded. The GitHub code API returned HTTP 401 and browser access required login. An independently accessible GitHub repository metadata query for carryless returned HTTP 200 with total_count: 0 and incomplete_results: false; this was a narrower metadata search, not exhaustive code search. The scope is bounded and carries no global novelty claim.