
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 7th, 2017, 09:15 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 255 Thanks: 23 Math Focus: Number theory  All prime Pythagorean triple
Does there exist a Pythagorean triple consisting only of prime numbers?

August 8th, 2017, 12:27 AM  #2 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 
Hint: At least one term of a Pythagorean triple must be even.

August 8th, 2017, 07:12 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,125 Thanks: 685  
August 8th, 2017, 09:11 AM  #4 
Senior Member Joined: May 2016 From: USA Posts: 750 Thanks: 302  $k,\ m,\ n \in \mathbb N^+.$ $(2k 1)^2 + (2m  1)^2 = (2n  1)^2 \implies$ $4k^2  4k + 1 + 4m^2  4m + 1 = 4n^2  4n + 1 \implies$ $1 + 1  1 = 1 = 4(n^2  n  k^2  m^2 + k + m) \implies$ $\dfrac{1}{4} = n^2  n  k^2  m^2 + k + m \implies$ $k \not \in \mathbb N^+ \text { or } m \not \in \mathbb N^+ \text { or } n \not \in \mathbb N^+.$ 
August 8th, 2017, 07:23 PM  #5  
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT  Quote:
 

Tags 
prime, pythagorean, triple, triples 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Prime software? not gimps or prime grid.  Kosxon  New Users  2  July 20th, 2017 03:08 PM 
Pythagorean prime in every prime twin  blind887  Algebra  2  April 12th, 2017 02:35 AM 
Special Pythagorean triple?  elim  Number Theory  6  October 17th, 2015 01:54 PM 
Formular for all Pythagorean Triple  M_B_S  Number Theory  24  November 21st, 2014 02:54 PM 
Pythagorean Triple and Rightangled triangles  Kiranpreet  Geometry  1  August 2nd, 2007 06:33 AM 