All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Trying to reach a Pythagorean triplet (Posted on 2017-02-12) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
solution Comment 1 of 1
(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

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information