Sum of the first few primes

Let $p_n$ be the $n$ . prime number. Suppose $S_{m-1}=k^2$ and $S_m=l^2$ , where $m>1$ and both of $k$ and $l$ are positive integers.

We can assume that $m>4$ . Since $p_m=S_m-S_{m-1}=(l-k)(l+k)$ $l-k=1$ and $k+k=p_m$ , and from that $l=\dfrac{p_m+1}{2}$ , so $S_m=\left (\dfrac{p_m+1}{2}\right )^2=S_m$ .

However this is a contradiction, since there aren't only prime numbers in the sum, so $S_m\leq(1+3+5+\dots+p_m)+2-1-9$

Therefore it si not possible.