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400 Notes on Explorations

Thomas, Rational tabulated values of trigonometric functions, Amer. Math.

Monthly 69 (1962), 789-793.

E.8. The problem of showing that f(z, y) is a polynomial if it is so in

each variable separately was twice posed in the American Mathematical

Monthly (# 4897 AMM 67 (1960), 295 & 68 (1961) 187; # E2940 AMM

89 (1982), 273 & 91 (1984), 142). A solution was published in the note,

F. W. Carroll, A polynomial in each variable separately is a polynomial,

Amer. Math. Monthly 68 (1961), 42.

E.9. If n = 1, then there are essentially three possibilities:

(1) the polynomial is constant and its range is a singleton;

(2) the polynomial is of odd positive degree and its range is all of R;

(3) the polynomial is of even degree and its range is a closed semi-infinite

interval of the form [m, oo) for positive leading coefficient or (00, m]

for negative leading coefficient.

The problem of determining the possible ranges of f(c, y) opened the
1969 Putnam Examination. To the possible surprise of the competitors
and perhaps even their supervisors, it turns out that the range can be an
open half line. The example given is (xy - 1)2 + x2. (See G.L. Alexander-
son, L.F. Klosinski & L.C. Larson, The W.L. Putnam Mathematical Com-
petition Problems and Sol&ions: 1965-1984 (MAA, 1985).) There are no
further possibilities when the number of variables exceeds 2. For complex
polynomials, the range is either a singleton or all of C. For polynomials
over Q defined on Q, the situation is complicated indeed.

E.lO. (a) (28, 53, 75, 84) and (65, 127, 248, 260) are instances of the
solution
(x3 + 1,223 - 1,x4 - 2x74 + x).
Other examples lead to the polynomial solutions

(3x2, 6x2 - 3x + 1, 32(3x2 - 22 + 1) - 1, 3x(3x2 - 2x + 1))

(3x2, 6x2 + 3x + 1, 3x(3x2 + 2x + l), 3x(3x2 + 2x + 1) + l),

one of which can be derived from the other by a change of variable x -
-x.
In his 1761 paper, Solutio generalis quorundam problematum diophante-
orum quae vulgo nonnisi solutiones speciales admittere videntur (Opera
Omnia (Series 1) 2, 428-458), Leonard Euler presents a number of formu-
lae for three cubes which add up to a fourth cube. One of these is

(x(x3 - y”), l/(x3 - y”), @x3 + I/“>, x(x3 + 2y3>>.