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## 'Generating Triples' printed from http://nrich.maths.org/

Charlie has been investigating square numbers. He decided to
organise his work in a table:

Charlie noticed some special relationships between certain square
numbers:

$$3^2+4^2=5^2$$ $$5^2+12^2=13^2$$

Sets of integers like ${3,4,5}$ and ${5,12,13}$ are called

Pythagorean Triples,
because they could be the lengths of the sides of a right-angled
triangle.

He wondered whether he could find any more...

Can you extend Charlie's table to find any more sets of Pythagorean
Triples where the hypotenuse is 1 unit longer than one of the other
sides?

Do you notice any patterns?

Can you make any predictions?

Can you find a formula that
generates Pythagorean Triples like Charlie's?
Can you prove that your formula
works?
Alison has been working on Pythagorean Triples where the
hypotenuse is 2 units longer than one of the other sides.

So far, she has found these:

$$4^2 + 3^2 = 5^2$$ $$6^2+8^2=10^2$$ $$8^2+15^2=17^2$$

Some of these are just scaled-up versions of Charlie's
triples, but some of them are new and can't be divided by a common
factor (these are called

primitive
triples).

Can you find more Pythagorean Triples like Alison's?

Can you find a formula for
generating Pythagorean Triples like Alison's?
Can you prove that your formula
works?
Here are some follow-up questions you might like to consider:

- Can you find Triples where the hypotenuse is 3 units longer
than one of the other sides? Or 4 units longer? Or...?
- Can you say anything about when such triples will be primitive
triples?

For a challenging extension
investigation, why not take a look at Few and Far
Between?