Reducing radicals in the spirit of Euclid. (May 2021)

• Record Type: Journal Article
• Authors: Girstmair, Kurt
• Is Part Of: Journal of symbolic computation. Volume 104(2021)
• Page Start: 356
• Page End: 365
• Publication Date: 2021-05
• Subjects: 12F10 -- 33F10; Radical extensions -- Zeilberger's algorithm; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.019
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrational y = d + R p, whose degree is 2 p, as a polynomial function of irrationals of degrees ≤ p . In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.