## Document Type

Report

## Source Publication Title

Technical Report 62

## Abstract

The existence is shown of infinitely many non-splitting perfect polynomials over GF(2d), GF(3d), GF(5d) for each odd integer d > 1, and over GF(2d) for each (even) integer d 1 0 (mod 3). Stronger results show that each unitary perfect polynomial over GF(q) determines an infinite equivalence class of unitary perfect polynomials over GF(q). The number SUP(q) of distinct equivalence classes of splitting unitary perfect polynomials over GF(q) is calculated for q = p and shown to be infinite for q # p. The number NSUP(q) of distinct equivalence classes of non-splitting unitary perfect polynomials over GF(q) remains undetermined, but is shown to be infinite whenever there are two relatively prime unitary perfect polynomials over GF(q) and one of them does not split. In particular NSUP(2d), NSUP(3d), and NSUP(5d) are infinite for each odd integer d > 1, and NSUP(2d) is infinite for each (even) integer d 1 0 (mod 3). Examples are given to establish NSUP(2) 33, NSUP(3) 16, and NSUP(5) 6. It is conjectured that for all primes p and odd integers d 1, NSUP(pd) is infinite.

## Disciplines

Mathematics | Physical Sciences and Mathematics

## Publication Date

5-1-1977

## Language

English

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Harbin, Mickie Sue; Bullock, A.T.; and Beard Jr., Jacob T. B., "Infinitely Many Perfect and Unitary Perfect Polynomials Over Some GF(q)" (1977). *Mathematics Technical Papers*. 81.

https://mavmatrix.uta.edu/math_technicalpapers/81