Re: Number Theory problem: seeking Calculus example

This is to add some pieces of information about nice
polynomials, and ask whether all of this information
is correct.

In this update, I use the list of references
of the main paper on nice polynomials:

Ralph Buchholz and James MacDougall,
When Newton met Diophantus:
A study of rational-derived polynomials
and their extension to quadratic fields
J. Number Theory 81 (2000), no. 2, 210-233.

I denote this paper by [BM].

The list of references in this paper is the most
comprehensive list available, and it is fantastic,
in the sense that we can re-find only a very small
part of it by using MathSciNet, Google,
and MathForum.

=========================================
ADDITIONAL INFORMATION 1
-----------------------------------------
A very useful online copy of this comprehensive
list of references is posted on the Web page
of Bill Dubuque, at the address:

http://groups.google.com/group/sci.math/msg/93257953e802e59
=========================================
ADDITIONAL INFORMATION 2
-----------------------------------------
To my knowledge, and according to what I see in the list
of references of [BM], the earliest publication on nice
polynomials is:

M. Chapple,
A cubic equation with rational roots such that it
and its derived equation also has rational roots,
Bull. Math. Teachers Secondary Schools 11 (1960), 5-7.
(Recently re-published in Aust. Senior Math. J. 4, No. 1 (1990), 57-60).

Does anyone know about any older publication on nice polynomials?
=========================================
ADDITIONAL INFORMATION 3
-----------------------------------------
The problem of the study of nice polynomials has been
mentioned in the following two lists of well known
unsolved problems:
-----------------------------------------
First list:
-----------------------------------------
Unsolved Problems Come of Age
Richard K. Guy
The American Mathematical Monthly, Vol. 96, No. 10,
Dec., 1989, pp. 903-909.
at the bottom of page 907 and top of page 908.

In this reference, it seems that a group of mathematicians
had found all nice quartics by applying the theory of
elliptic curves, but this information has to be checked.

Does anyone know whether this group of mathematicians
really completely solved the quartic case, and in what
publication they published a complete solution ?

-----------------------------------------
Second list:
-----------------------------------------
Richard Nowakowski,
Unsolved problems, 1969--1999,
Amer. Math. Monthly 106(10): 959--962, 1999.
=========================================
ADDITIONAL INFORMATION 4
-----------------------------------------
The term NICE POLYNOMIALS was independently coined
by the following authors of the following publications
that I found from the list of references of [BM]:
-----------------------------------------
1980:
T. Bruggeman and T. Gush,
Nice cubic polynomials for curve sketching,
Math. Mag. 53 (1980), 233-234.
-----------------------------------------
1990
Chris K. Caldwell,
Nice polynomials of degree 4,
Math. Spectrum 23, No. 22 (1990), 36-38.
-----------------------------------------
1990
Bill Galvin,
"Nice" cubic polynomials with "Nice" derivatives,
Austral. Senior Math. J. 4, No. 1 (1990), 17-21.
-----------------------------------------
1992
Jim Buddenhagen, Charles Ford, and Mike May,
Nice cubic polynomials, pythagorean triples, and the law of cosines,
Math. Mag. 65, No. 4, Oct. 1992, 244-249.
-----------------------------------------
Did I miss any other mathematician who coined the
term NICE POLYNOMIALS before these four publications
were published, or who coined it after the first one
was published, without knowing that is was already
coined ?
=========================================

With best regards,

Jean-Claude Evard
Western Kentucky University
Department of Mathematics
E-mail: Jean-Claude.Evard@xxxxxxx
.

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