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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