To tackle this problem from first principles provides a real
challenge. The solution published here was done by a 17 year old
who also proved the second part from first principles.
The arguments in the proofs use modulus arithmetic in an informal
Some students may like to read the articles on Pythagorean triples
first and then try to use similar arguments to prove the results in
this problem from first principles.
Another approach (and an easier one) would be to use the well known
formula for generating Pythagorean triples (proved in the articles
on Pythagorean triples) as the basis for a proof of the results in
If we can prove the result when $x^2 +y^2= z^2$ and $x, y$ and $z$
have no common factors (for primitive triples), are the results
then true for all right angled triangles?
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.