bibkey: berczes2010geometric
authors: Attila Bérczes
year: 2010
title: On the sumset of geometric progressions
doi: 10.5486/PMD.2010.4683
url: https://doi.org/10.5486/PMD.2010.4683
claim: Theorems 2.2 and 2.3 classify the diagonal and non-diagonal collisions for powers of the positive root of x³ − x − 1; Theorem 1.1 counts sumsets of consecutive powers.
strata_touched:
  - D5/S3/Arith/Additive/PlasticPowerCollisions
license: citation-only
triage: anchor

Plastic power collisions

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DOI: 10.5486/PMD.2010.4683

Source: https://doi.org/10.5486/PMD.2010.4683

Publisher PDF: https://publi.math.unideb.hu/paper/1503/download/

Theorem 1.1 is on pp. 262–263; Theorems 2.2 and 2.3 are on p. 264. The plastic-root cases are Theorem 2.3(1)–(3) with r = 1.

The publication is in Publicationes Mathematicae Debrecen 77/3–4, pp. 261–276. On p. 264, Theorem 2.3(1)–(3) gives the non-diagonal quadrinomials with exponent triples (4,3,2), (9,8,5), and (6,5,3), sharing a root with x⁷ − 2x⁵ + 1. The common non-unit root has minimal polynomial x³ − x − 1. Theorem 2.2, attributed to A. Schinzel, excludes two distinct diagonal trinomials with a common non-unit root.

For a real r > 1 with r³ = r + 1, the four normalized equal sums are 1 + r⁴ = r² + r³, 1 + r⁶ = r³ + r⁵, 1 + r⁷ = 2r⁵, and 1 + r⁹ = r⁵ + r⁸. Translation gives the collisions for arbitrary integer exponents, including negative exponents. The Lean proof uses the elementary integer difference equation in the repository source rather than formalizing the paper’s general polynomial classification.

The finite-subset application keeps sorted exponent pairs. Their gaps are 4 ↔ 1, 6 ↔ 2, 7 ↔ 0, and 9 ↔ 3. These distinct gaps prevent three unordered representations of the same real sum. Removing the outer representation of each present translated template leaves one representative of each sum. This application does not by itself count the templates in the particular nested family or establish its comparison with the golden-ratio power set.

No literature novelty is asserted for the collision classification.