bibkey: kagey2022a350532
authors: P. Kagey; OEIS Foundation Inc.
year: 2022
title: "OEIS A350532, sparse perfect powers over Z/2Z"
doi: null
url: https://oeis.org/A350532
claim: "Conjecture: T(2*n,2) = n. Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1."
strata_touched:
  - D5/S3/Factorization/Combinatorics/KageyA350532Binomials
  - D5/S3/Factorization/Combinatorics/KageyA350532Trinomials
license: citation-only
triage: anchor

Sparse perfect powers over GF(2)

Verified locator

DOI: null

Source: https://oeis.org/A350532

The entry is unpaginated. Its NAME is:

Triangle read by rows: T(n,k) is the number of degree-n polynomials over Z/2Z of the form f(x)^m for some m > 1 with exactly k nonzero terms; 1 <= k <= n + 1.

Its second and third conjectural COMMENT sentences are:

Conjecture: T(2*n,2) = n.

Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.

The count is of distinct nonzero polynomials, not power representations. Natural-number subtraction and division encode the third formula. The binomial formula includes n = 0. The trinomial formula is asserted for n >= 1.

The derivative obstruction and exponent contraction identify proper powers with two or three terms as squares. Support erasure then counts the degree-n roots by subsets of the n lower exponents. The shared counting formula also includes zero.

The source and bounded literature screen are specified by the binomial problem and the trinomial problem. Literature completeness is ASSUMED-UNVERIFIED; no exhaustive or originality claim is made.

The computation in the-omega-institute/trureturing-experiments, directory docs/reports/oeis-a350532-gf2-perfect-power-terms/, commit 9691967bdad8e9339e7fb9b1f5bea04dbead8c74, uses python3 check.py. Its recorded exit code is 0 with RESULT PASS; the SHA256 of check.py is 215183113f0d761af9b176c11fa95653a1ec36d94d82268aab1520190a34167d. These are the experiment readings quoted in the source issues, separate from the Lean proofs.