Choose an arbitrary triplet of positive integers.
Replace two integers by their sum and product, leaving the 3rd unchanged.
Lets call the above operation TPS.
Starting with (3,4,5) and applying continuously TPS is it possible to finally create a Pythagorean triplet?
Either prove the impossibility of doing it or show the chain of transformations creating such a triplet.
(1) A pythagorean triplet with two odd legs requires the square of the hypotenuse = 2 mod 4 which is impossible. So either one leg will be odd and one even with hyp odd or both legs and hyp will be even.
We start with (3,4,5), two odds and an even. Using TPS on the two odd numbers results in two evens and one odd. Then we're stuck since further TPS leaves the assignment of odd and even unchanged and by (1) that arrangement can never form a PT.
Using TPS on an odd and an even number leaves the assignment of odds and evens unchanged. Starting with (3,4,5) we have either (7,12,5) or (9,20,3). In both case the largest number is even. Further TPS keeps this even number even and doesn't alter that it is the largest of the three. Then the even number must eventually be the hypotenuse of any PT and by (1) that's impossible.
So it's not possible to begin with (3,4,5) and use TPS to create a PT.
Edited on February 13, 2017, 10:39 am
Posted by xdog
on 2017-02-13 10:36:36