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### Diophantine Equations

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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

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Linear Equations
High School Number Theory

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