# So many primes!

How many possibilities are there for the value of $$q$$, when:

$p_1^2 + p_2^2 + p_3^2 + \cdots +p_q^2 = c^{q-1}$

where $$p_1$$ to $$p_q$$ are any $$q$$ distinct primes, $$c$$ is a positive integer and $$q$$ is a prime of the form $$4k+3$$ (where $$k$$ is a positive integer)?

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