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Generating Triples

Age 14 to 16 Challenge Level:

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

 
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?