Prime Fibonacci prime > 3 expressed as the sum of two squares of distinct Fibonacci numbers

 
 
Can you prove that any prime Fibonacci prime > 3 can be expressed as the sum of two squares of distinct Fibonacci numbers?

 
the first few examples:
 

F_5 \; = \; 5 \; = \; 1^2 \; + \; 2^2 \; = \; F^2_2 \; + \; F^2_3

F_7 \; = \; 13 \; = \; 2^2 \; + \; 3^2 \; = \; F^2_3 \; + \; F^2_4

F_{11} \; = \; 89 \; = \; 5^2 \; + \; 8^2 \; = \; F^2_5 \; + \; F^2_6

F_{13} \; = \; 233 \; = \; 8^2 \; + \; 13^2 \; = \; F^2_6 \; + \; F^2_7

F_{17} \; = \; 1597 \; = \; 21^2 \; + \; 34^2 \; = \; F^2_8 \; + \; F^2_9

F_{23} \; = \; 28657 \; = \; 89^2 \; + \; 144^2 \; = \; F^2_{11} \; + \; F^2_{12}

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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