246 CHAPTER 7 NUMBER THEORY
infinitely many solutions. If m = 2(, and n = m + 1, then it is easy to check that
x = y = 2 t indeed satisfies (x^2 + l)m = (xy)n.
And these will be the only solutions. In other words, if m is odd, there are no
solutions, and if m is even, then there is the single solutionn = m + 1 ,x = y = 2m/2.
Problems and Exercises
7.4.5 Prove rigorously these two statements, which
were used in Example 7.4.3:
(a) If u 1. v and uv = x^2 , then u and v must be per
fect squares.
(b) If p is a prime and GCD(u, v) = p and uv =x^2 ,
then u = pr^2 , v = ps^2 , with r 1. s.
7.4.6 (Greece 1995) Find all positive integers n for
which -5^4 + 55 + 5" is a perfect square. Do the same
for 24 +2^7 +2".
7.4.7 (United Kingdom 1995) Find all triples of posi
tive integers (a,b, c) such that7.4.8 Show that there is exactly one integer n such that
28 + 2^11 + 2" is a perfect square.
7.4.9 Find the number of ordered pairs of positive in
tegers (x,y) that satisfy
xy
---n
x+y -.
7.4. 10 (USAMO 1979) Find all non-negative integral
solutions (nl,n 2 , ... ,nI 4 ) toni + n� + ... + ni 4 = 1,599.
7.4. 11 Find all positive integer solutions to abc - 2 =
a+b+c.
7.4.12 (Germany 1995) Find all pairs of nonnegative
integers (x,y) such that x 3 + 8x^2 - 6x + 8 = y 3.
7.4. 13 (India 1995) Find all positive integers x,y such
that 7 x - 3Y = 4.
7.4. 14 Develop a complete theory for the equation
x^2 + 2y^2 = z^2. Can you generalize this even further?7.4. 15 Twenty-three people, each with integral
weight, decide to play football, separating into two
teams of II people, plus a referee. To keep things fair,
the teams chosen must have equal total weight. It turns
out that no matter who is chosen to be the referee, this
can always be done. Prove that the 23 people must all
have the same weight.
7.4. 16 (India 1995) Find all positive integer solutions
x,y, z, p, with p a prime, of the equation xP + yP = pz.Pell's Equation
The quadratic diophantine equation x^2 - dy^2 = n,
where d and n are fixed, is called Pell's equation.
Problems 7 .4.17-7.4.22 will introduce you to a few
properties and applications of this interesting equation.
We will mostly restrict our attention to the cases where
n = ± I. For a fuller treatment of this subject, includ
ing the relationship between PeWs equation and con
tinued fractions, consult just about any number theory
textbook.
7.4. 17 Notice that if d is negative, then x^2 - dy^2 = n
has only finitely many solutions.
7.4. 18 Likewise, if d is perfect square, then x^2 -
dy^2 = n has only finitely many solutions.
7.4. 19 Consequently, the only "interesting" case is
when d is positive and not a perfect square. Let us
consider a concrete example: x^2 - 2y^2 = I.
(a) It is easy to see by inspection that (1,0) and
(3,2) are solutions. A bit more work yields the
next solution: (17, 12).
(b) Cover the next line so you can't read it! Now,
see if you can find a simple linear recurrence
that produces (3,2) from (1,0) and produces
(17,12) from (3,2). Use this to produce a new
solution, and check to see if it works.