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Nice polynomials with three roots

Published online by Cambridge University Press:  01 August 2016

Jonathan Groves
Jonathan Groves*
Affiliation:
Department of Mathematics, Patterson Office Tower 713, University of Kentucky, Lexington, KY 40506-0027 USA, e-mails: JGroves@ms.uky.edu, Jonny77889@yahoo.com
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Nice polynomials are polynomials whose coefficients, roots, and critical points are integers. The earliest papers on nice polynomials [1, 2, 3] give an explicit formula for all nice cubics. All the derivations of this formula use Pythagorean triples.

In 1999 the problem of finding, constructing, and classifying nice polynomials was added to the list of unsolved problems [4] in The American Mathematical Monthly. Other papers soon followed, including [5] and [6], with extending the results of nice polynomials to polynomials with coefficients, roots, and critical points in integral domains D. Such polynomials are called D-nice.

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Articles
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The Mathematical Gazette , Volume 92 , Issue 523 , March 2008, pp. 1 - 7
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Copyright © The Mathematical Association 2008

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References

1. Bruggeman, Tom and Gush, Tom, Nice cubic polynomials for curve sketching, Math Magazine 53(4) (1980) pp. 233–234.Google Scholar
2. Chapple, M., A cubic equation with rational roots such that it and its derived equation also has rational roots, Bull. Math. Teachers Secondary Schools, 11 (1960) pp. 5–7 (recently republished in Aust. Senior Math. J. 4(1) 1990 pp. 57–60).Google Scholar
3. Zuser, Karl, Uber eine gewisse Klasse von ganzen rationalen Funktionen 3. Grades (in German), Eiern. Math, 18 (1963) pp. 101–104.Google Scholar
4. Nowakowski, Richard, Unsolved problems, 1969–1999, Amer. Math. Monthly 106(10) (1999) pp. 959–962.Google Scholar
5. Buchholz, Ralph H. and MacDougall, James A., When Newton met Diophantus: a study of rational-derived polynomials and their extensions to quadratic fields, J. Number Theory 81 (2000) pp. 210–233.Google Scholar
6. Evard, Jean-Claude, Polynomials whose roots and critical points are integers, posted on the Website of Arxiv Organization at the address http://arxiv.org/abs/math/0407256.Google Scholar