Some Variations On Euclid's Method
A trivial sequence that generates some but not all possible triples is based on the positive integers starting with a = 3.
- If a is odd, then b = a2/2 − 1/2 and c = b + 1
- If a is even, then b = a2/4 − 1 and c = b + 2
Note that a = 3 and a = 4 produce the same primitive triple and that when a = 4n + 2 for integer n (thus a is even), then the triple produced is non-primitive and simply the corresponding primitive from the odd sequence with all values doubled.
More formally: given a positive integer n, the triple can be generated by the following two procedures: (see http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html )
Example: When n = 2, the triple produced is 5, 12, and 13. (This formula is actually the same as method I, substituting m with 2n + 1.)
Alternatively, one can generate triples from even integers using the following formulas. Given that m is a positive even number,
Example: When m = 4, the triple produced is 8, 15, and 17 (this formula is another specific case of method I, substituting n with 1).
These two methods generally produce different results, but when n is 1 in the first formula and m is 2 in the second formula, the result is 3, 4, 5. This method yields Pythagorean triples (not all of them primitive) for any given value, under the conditions given, but it does not yield all valid Pythagorean triples, or even all primitive Pythagorean triples. The pattern that the odd values express is visible in the scatter plot of triples as the darker radiating lines. The even sequence is present, but harder to see. The smallest primitive triple based on values for c that this sequence does not find is (20, 21, 29) whose multiples appear as a pair of lighter radiating lines near the diagonal of the scatter plot.
Given the integers u and v, (see: http://www.math.rutgers.edu/~erowland/pythagoreantriples.html )
Example: For u = 3 and v = 5, a = 39, b = 80, c = 89. (This formula is actually the same as method I, substituting m and n with u + v and v.)
For the resulting triple to be primitive, u and v must be co-prime and u must be odd.
A particularly elegant version of this method is to calculate
Then
Read more about this topic: Formulas For Generating Pythagorean Triples
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