How many integers $a,b,c$, both positive and negative, such that $P=a^b b^c c^a$ is a prime number? 
How many integers $a,b,c$, both positive and negative, such that $P=a^b b^c c^a$ is a prime number ?

If $a,b,c$ are positive, then two of $a,b,c$ equal to $1$. Assume that $b=c=1$, then $a$ is any prime number.
WLOG, if $c<0$ and $a,b>0$, then $a$ must be even, so $a=2$, thus $b=1$ and $c=-1$
If $a,b,c<0$, then $P$ is smaller than $1$, so $P$ won't be a prime.
However if only one of $a,b,c$ is positive, how can we find $a,b,c$ ? 

WLOG assume $b,c<0$, $a>0$. Then $a^b$ and $b^c$ will be fractional unless $a=1$, $b=-1$. So we need to find $1\cdot (-1)^c \cdot c=p$ for $c<0$. Since $c$ is negative, it must also be odd so that $(-1)^c$ will be negative and the entire product will be positive. Hence, $c$ can be the negative of any odd prime.
