#### Article Title

#### Document Type

Research

#### Abstract

If 4X = 4(x^{P}-1)/(x-1) where p is an odd prime then 4X = Y^{2}-(-1)^{(P-1)/2 }pZ^{2}, 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

© Copyright 1936 by the 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