Diophantine equations

problem

Upsetting Pythagoras

Age

14 to 18

Challenge level

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2
article
Euclid's Algorithm I

How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with the solutions x and y being integers? Read this article to find out.
article
Euclid's Algorithm II

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
problem

Shades of Fermat's Last Theorem

Age

16 to 18

Challenge level

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?
problem

BT... Eat Your Heart Out

Age

16 to 18

Challenge level

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?