Is the following an existing conjecture or a conjecture at all?

Question

the following floated to my mind today, can you verify if it stands to be true, or is a pre-existing conjecture. If not, can you correct me? And if it is one, can you prove it?

Prime factorisation of any perfect square $P$ of the form

$$P=a_1^{x_1}\times a_2^{x_2}\times a_3^{x_3}\times.........\times a_n^{x_n} $$ where $$x_1,x_2≠0$$ which implies that atleast two non unity factors exist,then $$a_1^{x_1}+ a_2^{x_2}+ a_3^{x_3}+.........+ a_n^{x_n}=Z$$ where $Z$ is perfect square, is only applicable for $P=144$.

Example: for $P=144$, prime factorisation is as follows: $$2^4\times 3^2$$ and $$2^4+ 3^2=25=5^2$$

Answer 1

There are many solutions.

This PARI/GP program searches for solutions :

? f(n)={w=factor(n);x=component(w,1);y=component(w,2);sum(j=1,length(x),x[j]^y[j])}
%80 = (n)->w=factor(n);x=component(w,1);y=component(w,2);sum(j=1,length(x),x[j]^y[j])
? for(m=1,2*10^4,s=m^2;if(omega(s)>1,if(issquare(f(s))==1,print(m^2,"  ",f(s)))))
144  25
4601025  324
7452900  256
16769025  324
18533025  1764
42837025  484
81540900  1936
111197025  1764
142444225  1156
165636900  400
179506404  1024
184552225  676
246741264  484
373262400  676
387893025  10404
?