Complex solutions for Fermat-Catalan conjecture The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + \frac{1}{n} +\frac{1}{k} <1$. There are currently only 10 solutions known (listed at the end of this post). My question concerns the case where $a, b, c$ are positive coprime Gaussian integers. I've found two solutions. Is there a clever method for finding more? I've used brute force techniques.


*

*$(8+5i)^2+(5+3i)^3=(1+2i)^7$  

*$(20+9i)^2+(1+8i)^3=(1+i)^{15}$

*$(1+2i)^7+(49+306i)^2=(27+37i)^3$ (Zander)

*$(44+83i)^2+(31+39i)^3=(5+2i)^7$ (Zander)

*$(238+72i)^3+(7+6i)^8=(7347−1240i)^2$ (Oleg567)


Here are the known solutions over integers.


*

*$1^m+2^3=3^2$ 

*$2^5+7^2=3^4$

*$13^2+7^3=2^9$

*$2^7+17^3=71^2$

*$3^5+11^4=122^2$

*$33^8+1549034^2=15613^3$

*$1414^3+2213459^2=65^7$

*$9262^3+15312283^2=113^7$

*$17^7+76271^3=21063928^2$

*$43^8+96222^3=30042907^2$


I found the next complex solution! :)
$$(238+72i)^3+(7+6i)^8=(7347−1240i)^2$$
(There is no new method here. Just small contribution to the problem.)
