## Diophantine Equations

Named after *Diophantus of Alexandria* ( ca. 250 A.D. ), who in his book *Arithmetica*
seeks to solve linear and square equations and especially to find their integer solutions.

The program computes the integral solutions of the equation a·x - b = m·y with m>0.

This for example permits the determination of the integral points in a straight line.

### Example:

The straight line with the equation y = 7/3·x - 5/3
⇔ 7·x - 3·y - 5 = 0 ; x,y integer

comprises the integer points

L = { (x|y) | x=2+3t, y=3+7t and t integer }
= { (2|3),(5|10),(-1|-4),(8|17), ... }

### Complement:

The Diophantine equation of the 2nd degree x^{2} + y^{2} = z^{2} leads to the Pythagorean number triplet.

In the famous "Fermat's Last Theorem" he claimed that the equation x^{n} + y^{n} = z^{n}
has no integral solutions for n>2.

The proof of this theorem has occupied mathematics for 400 years and was only achieved in 1995 by the English mathematician Andrew Wiles.
Simon Singh describes the long way until then in his book FERMAT'S LAST THEOREM (ISBN 978-1857025217), which shows in an excellent way
the difference between mathematics and mere arithmetic or problem-solving.

### See also:

Wikipedia: Diophantine Equations