 
Summary: Diophantine Equations I
Putnam practice
October 27, 2004
In his book Arithmetica Diophantus discussed the problem of finding
all integral (or rational) solutions of an equation in two or more variables.
Fourteen centuries later Fermat was reading Diophantus' book and asked
the famous question: show that for n > 2
xn
+ yn
= zn
has no solution, where x, y, z are rational numbers. This question was an
swered only recently by A. Wiles.
There is no general algorithm for dealing with Diophantine equations.
However, there are some techniques that work in particular cases.
Factoring
Problem 1 Find all pairs (x, y) of integers that satisfy the equation
x(y + 1)2
= 243y
Congruence argument
Problem 2 Prove that x2 = 3y2 + 8 has no solution in integers x, y.
