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Finding such a set of "a"'s with prime factors raised to an even power, is pretty non-trivial, and is not performed by the sieving clients or servers.
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Initially wee derive a network in which nodes are primes and edges are drawn between
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the two large primes in a double-partial relation, or between 1 and the single prime of a partial relation.
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Then we run around looking for cycles in the network, which represent large primes that appear as squares when the two corresponding residues are multiplied together.
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Multiply all relations together, divide by all squared large primes, and you turn lots of partials and double partials into full relations.
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