Advanced book on Mathematics Olympiad

276 5 Number Theory

Example.The numbers 2FnFn+ 1 ,Fn− 1 Fn+ 2 andF 2 n+ 1 form a Pythagorean triple.

Solution.In our parametrization, it is natural to tryu=Fn+ 1 andv=Fn. And indeed,

u^2 −v^2 =(u−v)(u+v)=(Fn+ 1 −Fn)(Fn+ 1 +Fn)=Fn− 1 Fn+ 2 ,

while the identity

F 2 n+ 1 =u^2 +v^2 =Fn^2 + 1 +Fn^2

was established in Section 2.3.1. This proves our claim. 

801.Given that the sides of a right triangle are coprime integers and the sum of the legs

is a perfect square, show that the sum of the cubes of the legs can be written as the

sum of two perfect squares.

802.Find all positive integersx, y, zsatisfying the equation 3x+y^2 = 5 z.

803.Show that for no positive integersxandycan 2x+ 25 ybe a perfect square.

804.Solve the following equation in positive integers:

x^2 +y^2 = 1997 (x−y).

5.3.3 Pell’s Equation

Euler, after reading Wallis’Opera Mathematica, mistakenly attributed the first serious

study of nontrivial solutions to the equation

x^2 −Dy^2 = 1

to John Pell. However, there is no evidence that Pell, who taught at the University of
Amsterdam, had ever considered solving such an equation. It should more aptly be called
Fermat’s equation, since it was Fermat who first investigated it. Nevertheless, equations
of Pell type can be traced back to the Greeks. Theon of Smyrna used the ratioxyto
approximate

√

2, wherexandyare solutions tox^2 − 2 y^2 =1. A more famous equation

is Archimedes’problema bovinum(cattle problem) posed as a challenge to Apollonius,

which received a complete solution only in the twentieth century.

Indian mathematicians of the sixth century devised a method for finding solutions to

Pell’s equation. But the general solution was first explained by Lagrange in a series of

papers presented to the Berlin Academy between 1768 and 1770.

Lagrange’s theorem.IfDis a positive integer that is not a perfect square, then the

equation