Diophantine equations

There are 25 NRICH Mathematical resources connected to Diophantine equations
Upsetting Pitagoras
problem

Upsetting Pitagoras

Age
14 to 18
Challenge level
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Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2
Shades of Fermat's Last Theorem
problem

Shades of Fermat's Last Theorem

Age
16 to 18
Challenge level
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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?
Euclid's Algorithm II
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.
Euclid's Algorithm I
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.
Why stop at Three by One
article

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.