Diophantine Equations

Date: 11/17/97 at 15:37:25
From: Robert Felgate
Subject: Diaphantine equations

Dear Dr. Math,

We have been investigating Pythagoras' theorem, and our teacher has 
suggested we search the web for information about diaphantine 
equations. We have not been very successful and wonder if you would 
help us.

Thank you.
Robert and his Mum

Date: 11/17/97 at 16:50:59
From: Doctor Rob
Subject: Re: Diaphantine equations

Probably it would help if you spelled the word "Diophantine" - the 
equations are named after Diophantus of Alexandria, a famous Greek 
mathematician of antiquity.

A Diophantine equation is one equation in at least two variables, say 
x and y, whose solutions (x,y) are required to be whole numbers 
(integers). Some such equations have no solutions. Some have a finite 
number. Some have infinitely many. The example of the Pythagorean 
equation is one:

   a^2 + b^2 = c^2.

If you are asked for integer solutions, this is a Diophantine 
equation. You already know some solutions, I am sure:  

(a,b,c) = (0,0,0), (0,1,1) (0,1,-1), (0,-1,1), (0,-1,-1), (1,0,1), 
(1,0,-1), (-1,0,1), (-1,0,-1),

for example. You can see that for this equation, you can assume that
a, b, and c are all positive. From that solution you can obtain others
by changing signs of some or all of a, b, and c. Furthermore, you can
assume a <= b, since you can take any solution and swap a and b to get
another. In this case there are infinitely many solutions.

Another Diophantine equation might be 5*x + 7*y = 57. This, too, has 
an infinite number of solutions, and they are given by x = 3 + 7*t,
y = 6 - 5*t, where t is any integer.

This kind of equation is studied when you take Number Theory as an
advanced undergraduate or beginning graduate student.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/