Diophantine Equations
Just the integers, please.
Diophantus of Alexandria
Can we solve an equation using only integers?
These are the most popular such equations to solve:
Each of them is like trying to solve a puzzle. In fact many can be solved by trial and error, either with pencil and paper or using a computer.
Linear Diophantine Equations
Are of this type:
ax + by = c
They can sometimes be solved.
See Linear Diophantine Equations to learn more.
Pythagorean Triples
Pythagorean Triples are the integer solutions to the famous Pythagoras' Theorem x2 + y2 = z2
Example: the 3, 4, 5 Triangle
32 + 42 = 52
So 3, 4 and 5 are integer solutions to x2 + y2 = z2
Can you find other integer solutions? The page Pythagorean Triples talks all about them. But try to find some yourself first!
Powers of 3 or more
Can we solve the same equation but with powers of 3 or more?
x3 + y3 = z3 ?
xn + yn = zn (for n>2) ?
Back in 1637 Fermat thought it was impossible and it became known as Fermat's Last Theorem. It was finally proved in 1995 by Andrew Wiles. Wow.
So there are no positive integer solutions to xn + yn = zn for n>2.
Taxicab Numbers
Ramanujan
A story by GH Hardy about a visit to the famous Ramanujan:
I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one.... "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
Which can be written:
w3 + x3 = y3 + z3
He was indeed right, because:
13 + 123 = 93 + 103 = 1729
The next Taxicab Number is much higher!
Other Types
There are many other types of Diophantine Equations and they are all interesting journeys in their own way.