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
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?
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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.
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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.
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Why stop at three by one

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.