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August 7th, 2017, 10:15 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 303 Thanks: 25 Math Focus: Number theory  All prime Pythagorean triple
Does there exist a Pythagorean triple consisting only of prime numbers?

August 8th, 2017, 01:27 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, 08:12 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,672 Thanks: 741  
August 8th, 2017, 10:11 AM  #4 
Senior Member Joined: May 2016 From: USA Posts: 903 Thanks: 359  $k,\ m,\ n \in \mathbb N^+.$ $(2k 1)^2 + (2m  1)^2 = (2n  1)^2 \implies$ $4k^2  4k + 1 + 4m^2  4m + 1 = 4n^2  4n + 1 \implies$ $1 + 1  1 = 1 = 4(n^2  n  k^2  m^2 + k + m) \implies$ $\dfrac{1}{4} = n^2  n  k^2  m^2 + k + m \implies$ $k \not \in \mathbb N^+ \text { or } m \not \in \mathbb N^+ \text { or } n \not \in \mathbb N^+.$ 
August 8th, 2017, 08:23 PM  #5  
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT  Quote:
 

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prime, pythagorean, triple, triples 
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