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Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Data Chunks

Age 14 to 16
Challenge Level
Systematic working and recording of results help a lot here.

Conjectures are important, and should be encouraged, but along with a challenge to really explain why any claim might be true generally.

We hope the problem will give students a genuine pleasure in discerning real structure, and lead their interest on into Number Theory.

The articles on the NRICH site (see link from problem page) are an excellent follow-on.