EMAT 6700






The Babylonians had written thousand of Pythagorean triples some with extremely large numbers.  They must have had some way of generating them other than multiplying the 3,4,5 triple by a constant.

Here is a one way they might have used to find a Pythagorean triple.

First chose a prime number say 17.

17² = 289
divide 289 by 2 = 144.5
round 144.5 down to 144
round 144.5 up to 145

The new Pythagorean Triple is 17, 144, 145.








Here is an algebraic proof of the above:

x² + y² = z²
x² = z² - y²
1·x² = (z - y)(z + y)
so let (z - y) = 1 then (z + y) = x²
z= y + 1
by substitution 2y + 1 = x²
y = x²/2 - 1/2
(notes from a Math 3200 presentation)

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