1.3. Complex Numbers 13E.5. Polynomials, some of whose values are squares. The square
integers 1 = 12, 25 = 52 and 49 = 72 are in arithmetic progression. This
means that there is a linear polynomial, for example 1 + 24t, whose values
are squares for three consecutive integer values oft. Is it possible to find
a linear polynomial which takes a square value at four consecutive integer
values of the variable t?
If a quadratic polynomial p(t) is the square, [q(t)j2 of a linear polynomial
with integer coefficients, then it will always assume square values for integer
values of t. Is the converse true? If not, what is the maximum number
of square values which a quadratic polynomial (not equal to the square
of a linear polynomial) might assume at consecutive integer values of its
variable. In particular, determine a quadratic polynomial which assumes
six consecutive square values.
Somewhat related to these questions is that of taking two finite disjoint
subsets U and V of the integers and seeing whether there exists a poly-
nomial p(t) with integer coefficients which is square when t belongs to U
and nonsquare when t belongs to V. For example, one can find a quadratic
polynomial f(t) for which f(l), f(9), f(8), f(6) are squares but f(1986) is
a nonsquare.1.3 Complex Numbers
The roots of the quadratic equation t2 $- 2t + 8 = 0 are -1 + J-‘;i and
-1 - fl. Thus, even simple polynomial equations lead us beyond the real
number system. If we expand the system to include such “imaginaries” as
fl, we shall see that the theory of polynomials can be placed in a very
natural setting indeed. A complex number is one which can be written in
the form z + yi where x and y are real and i2 = -1. Do not worry about
what i “means”; all we need to know is that its square is -1. The set C
of all complex numbers x + yi can be represented by points (x, y) in the
Cartesian plane; such a representation is called the Argand diagram and
we refer to the complex plane. The x-axis is called the real axis and the
y-axis the imaginary axis.
In discussing complex numbers, it is useful to have some more terminol-
ogy:
Let z = z + yi denote a complex number, with x and y real.
The real part, Re z, of z is the number 2.
The imaginary part, Im z, of z is the number y.
The complex conjugate, IF, of z is x - yi.
The modulus or absolute value of z, denoted by IzI, is dw. This is
the distance from the origin to the point representing z.
The argument of z is the angle between the real axis and the line joining
0 and z, measured in the counterclockwise direction. It is denoted by arg z.
Generally, we assign to arg z a value between 0 and 27r.