Unknown

30 1. Fundamentals

is satisfied by (A, B, C, D) =

(6,23,32, w, (16,87,122,149),

(39,70,91,108), (51,148,203,246),

(59,228,317,386), (59,630,889,1088),

(79,242,333,404), (83,516,725,886),

(108,157,194,225), (147,302,401,480),

(225,296,353,402), (324,557,718,849),

(402,499,580,651).

There is a family of solutions in which A, B, C, D are given by evaluating

linear polynomials at integer values. However, the numerical data above

will suggest polynomial solutions of higher degree.

(c) Let r be a fixed numerical parameter. Find polynomial solutions to

X2+rXY+Y2=Z2.

(d) Show that there are infinitely many integers which are equal to the
sum of the squares of their digits written to some base.
(e) Show that there are infinitely many integers which are equal to the
sum of the cubes of their digits written to some base. For example, 17
written in base 3 has the representation (122)3 and is the sum of the cubes
of 1, 2 and 2.
(f) In Exploration E.2, we considered pairs of sets of numbers for which
the sum of various powers of the elements of one were equal to the cor-
responding powers of the elements of the other. Look for pairs of sets of
polynomials which have the same property.

1.6 Basic Number Theory and Modular

Arithmetic

What numbers can be expressed as the difference of two integer squares?
Since (x + 1)2 - x2 = 22 + 1, it is clear that every odd number can be so
expressed. How about 98? If x2 - y2 = 98, then 2 and y must be either
both even or both odd. But in this case, it is straightforward to argue that
x2 - y” is divisible by 4. Thus, the representation of 98 is not possible.
This type of argument occurs frequently in studying polynomials with
rational and integer coefficients. Accordingly, in this section we will re-
view some basic properties of the number system. Another reason for the
importance of knowledge about the structure of integers is the fact that
the family of polynomials shares much of this structure and the theory is
developed in an analogous way.
First, some terminology. N denotes the set {1,2,3,.. .} of natural num-
bersandZtheset{.... -2,-1,0,1,2,.. .} of all integers. For any pair a, b,
of integers, we say that a divides b (in symbols: ulb) if and only if there is