Let $n$ satisfy the equation, $a^n+b^n=c^2$. When $a,b,c$ are prime numbers except $n$, there is one solution at least for the equation.
How to find the biggest possible value of $n$?
Let $n$ satisfy the equation, $a^n+b^n=c^2$. When $a,b,c$ are prime numbers except $n$, there is one solution at least for the equation.
How to find the biggest possible value of $n$?
I'm not sure I completely understand the "a,b,c, are prime numbers on one solution at least", but $\,2^n+2^n=2\cdot 2^n=2^{n+1}\,\Longrightarrow \,$ for any odd natural $\,n\,$: $\,2^n+2^n=2^{n+1}=\left(c^{\frac{n+1}{2}}\right)^2\,\,,\,with\,\,\, c:=2^{\frac{n+1}{2}}$
For example, $\,2^7+2^7=2^8=(2^4)^2$
Thus, there is no greatest value of n.
Henri Darmon and Loic Merel, Winding quotients and some variants of Fermat's last theorem, J. Reine Angew. Math. 490 (1997), 81–100, MR1468926 (98h:11076) settled the problem. According to the review by Ken Kramer, they show that when $n\ge4$ the equation $x^n+y^n=z^2$ has no solution in integers with $|xyz|\gt1$ and $\gcd(x,y,z)=1$.
What follows is my earlier answer, which may now be ignored --- I leave it up solely because there were some comments on it.
The biggest $n$ is probably 3.
There are solutions (in coprime integers, here and below) to $a^3+b^3=c^2$.
There are no solutions to $a^4+b^4=c^2$.
There are no known solutions to $a^n+b^n=c^2$, for any $n\gt3$. I am not sure whether it has been proved that no solutions exist.
It occurs to me that with all the conditions on the variables there is an elementary solution to this problem, and my earlier solution quoting Darmon and Merel is overkill.
First, if $a,b,c$ are all to be prime numbers, then it must be the case that all three are even, or that exactly one of them is even. In the first case, we have $2^n+2^n=2^2$ and the sole solution is $n=1$. So now we may suppose exactly one of $a,b,c$ is even. If $c$ is even, we have $a^n+b^n=4$ and the only solution is $1^1+3^1=4$, so again $n=1$. So now we may assume $a$ is even, and we have $2^n+p^n=q^2\tag0$ where $p,q$ are odd primes, and $n$ is not prime. If $n$ is odd then we have $(2+p)(2^{n-1}-2^{n-2}p+\cdots+p^{n-1})=q^2\tag1$ Since $q$ is prime, the only factorizations of $q^2$ are as $1\times q^2$ and $q\times q$. But for $n\gt1$, the second term on the left in (1) is greater than the first term, which is greater than 1, so (1) is impossible. Thus, in this case also $n=1$, and there are solutions such as $2+7=3^2,\quad2+23=5^2,\quad2+47=7^2$ and so on. Finally, we suppose $n=2k$ is even in (0). Then $2^{2k}=q^2-(p^k)^2=(q+p^k)(q-p^k)$ So $q+p^k$ and $q-p^k$ must both be even and must both be powers of 2, but their sum, $2q$, is not a multiple of 4, so we must have $q+p^k=2^{2k-1},\quad q-p^k=2$ This implies $p^k=2^{2k-2}-1=(2^{k-1}+1)(2^{k-1}-1)$ So $2^{k-1}+1$ and $2^{k-1}-1$ are both powers of $p$, but their difference, 2, is not a multiple of $p$, so $p^k=2^{k-1}+1,\quad1=2^{k-1}-1$ The second equation gives $k=2$, then the first gives $p^2=3$, contradiction.
So, we have an utterly elementary proof that $n=1$.