## 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:

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

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

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

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

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

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