I'm looking for numbers of the form $$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$ where $p_{i}$ are prime numbers, $p_{i}\ne p_{j}$ and $\alpha_{_{k}}\in\mathbb{N}$
The first exemple is $$(2^2)^2+(3^1)^2=(5^1)^2$$ I did a quick look at pythagorean triplets but could't find any. So, I wonder if this is the only exemple or if there are finitely more or infinitely many.
What is known about this?