Maths Item of the Month – February 2012
How odd is Pythagoras?
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Pythagorean triples are sets of positive integers (a, b, c) where a + b = c . A Pythagorean
triple is primitive if there isn't a common factor that divides a, b and c.
(3, 4, 5) and (5, 12, 13) are primitive Pythagorean triples but (6, 8, 10) is not as 2 divides 6,
8 and 10.
Is the smallest number in a primitive Pythagorean triple always odd?
Is the largest number in a primitive Pythagorean triple always odd?
Solution
To solve this problem we will use Euclid’s Formula. Euclid’s formula says that all Primitive
Pythagorean triples can be described in the form
a = m2 – n2, b = 2mn, and c = m2 + n2,
where m and n are positive integers, and m > n.
It follows that that the middle number, b (or 2mn) is always even. It is not possible for the
other two numbers to both be even as well, as then the triple would not be primitive; it would
be divisible by 2.
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It also follows that if m – n is odd, so too must m + n be odd.
Therefore, for a Primitive Pythagorean triple, a must be odd, b must be even, and c must be
odd.
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08/02/13 © MEI