Diophantine Equations

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

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

This can be used, for example, to determine the integer points on 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  x² + y² = z²  leads to the Pythagorean number triplet.

In the famous "Fermat's Last Theorem" he claimed that the equation  xⁿ + yⁿ = zⁿ  has no integer solutions for n>2.

The proof of this theorem has occupied mathematicians 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