The Decomposition of 4(x^p-1)/(x-1)

Authors

Cornelius Gouwens, Iowa State CollegeFollow

Document Type

Research

Abstract

If 4X = 4(x^P-1)/(x-1) where p is an odd prime then 4X = Y²-(-1)^(P-1)/2 pZ², where Y and Z are polynomials in x with integral coefficients. For p = 37 we find the decomposition cited in "Recherches sur la theorie des nombres" by M. Kraitchik (1924) p. 126. For 37 ≤ p ≤ 61 the decomposition is given by Pocklington in "Nature," VoL 107 (1921) pp. 456 and 587. For 67 ≤ p ≤ 97 the results are given by Gouwens in "The Mathematical Monthly, Vol. 43, (1936) page 283. Herewith are presented the results for 101 ≤ p ≤ 199. For all decompositions Y is a polynomial of degree (p-1)/2 and Z is a polynomial of degree (p-3)/2. Y is listed first in each case, then Z.

Publication Date

1936

Journal Title

Proceedings of the Iowa Academy of Science

Volume

43

Issue

1

First Page

255

Last Page

262

Copyright

©1936 Iowa Academy of Science, Inc.

Language

en

File Format

application/pdf

Recommended Citation

Gouwens, Cornelius (1936) "The Decomposition of 4(x^p-1)/(x-1)," Proceedings of the Iowa Academy of Science, 43(1), 255-262.
Available at: https://scholarworks.uni.edu/pias/vol43/iss1/74