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March 5th, 2009, 06:18 AM  #1 
Member Joined: Nov 2006 Posts: 54 Thanks: 0  A Harmonic And Near Pythagorean Puzzle
Determine all possible triplet(s) (x,y,z) of positive integers, with x< y< z, such that: x,y and z (in this order) are in harmonic sequence, and: x^2 + y^2 = z^2  11 Note: x, y and z (in this order) are in harmonic progression (or, harmonic sequence), if 1/x, 1/y and 1/z(in this order) are in arithmetic progression (or, arithmetic sequence). 
March 5th, 2009, 08:01 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Harmonic And Near Pythagorean Puzzle Quote:
 
March 5th, 2009, 08:10 AM  #3  
Member Joined: Nov 2006 Posts: 54 Thanks: 0  Re: A Harmonic And Near Pythagorean Puzzle Quote:
x, y and z are in harmonic progression (or, harmonic sequence), if 1/x, 1/y and 1/z are in arithmetic progression (or, arithmetic sequence). So, y is the harmonic mean of x and z. I confirm having amended the original problem text to make the intent clearer.  
March 5th, 2009, 08:49 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Harmonic And Near Pythagorean Puzzle
(3, 4, 6) was the only solution I could find. There aren't any other small solutions.


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