Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.
Source: China Zhejiang Fuyang , 28 Jul 2014
Tags: number theory unsolved, number theory
Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.