Start off your new year with primes!

Level pending

Suppose, p, q and r are prime positive integers, satisfying pq = r + 1 and 2(p^{2} + q^{2}) = r^{2} + 1. Let there be n ORDERED solutions to (p, q, r) i.e. (p{1}, q{1}, r{1}), (p{2}, q{2}, r{2}),...,(p{n}, q{n}, r{n}). Find \sum{i=1}^n (p{i} + q{i} + r_{i}).

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