| START w/ Odd #s | START w/ EVEN # | ALL EVEN #s | EVEN Odd Odd |
| 3-4-5 | 12-13-16 | 6-8-10 | 6-15-17 |
| 5-12-13 | | 10-24-26 | 20-21-29 |
| 7-24-25 | | 16-30-34 | 16-63-65 |
| 9-12-15 | | 12-16-20 | 72-135-153 |
| 9-40-41 | | | 24-143-145 |
| 3-4-5 | 3-4-5 | 6-8-10 | 8-15-17 | 5-12-13 |
| 3(2)-4(2)-5(2) | 3(3)-4(3)-5(3) | 6(2)-8(2)-10(2) | 8(2)-15(2)-17(2) | 5(2)-12(2)-13(2) |
| 6-8-10 | 9-12-15 | 12-16-20 | 16-30-34 | 10-24-26 |
I noticed that it is rare to have a triple with the pattern of even-odd-even, but there are a lot of triples with the pattern of odd-even-odd.
I also noticed that in the triples 3-4-5 & 9-12-15, in the first group, if you multiplied all the numbers by 3 you would get the numbers of the second group of Triples.
There is another pattern similar with two Triples in the all even column (shown below). if you multiply all the numbers in 6-8-10 by 2 you will get 12-16-20 which is also a Pythagorean Triple.
Another thing that I observed was about the Pythagorean Triple 8-15-17. if you multiply the numbers by 2 the product is 16-30-34 which is another Pythagorean Triple that I found using the formula of Diophantus.
When you multiply the numbers in the Pythagorean Triple, 5-12-13 by 2 then the product is the Triple, 10-24-26.
One thing that stood out to me was that the Triples that involve odd numbers. When I multiplied them by 2 they turned out to be Triples that have all even numbers in them, that happened with three sets of Pythagorean Triples which were, 3-4-5, 8-15-17, and 5-12-13.
Do you think that this pattern continues for later numbers - that is to say, are there always more triples that start with odds, or do evens start to overtake? Is it possible to have more than one triple that start with the same number? Why or why not? What other questions do you want to investigate? What here seems interesting to you?
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