The client is using some sort of sieve to mine for primes. Is this so that you can check for chains starting at multiples of the origin using the same sieve?
When you hash to produce your origin why not just check for the chains using the verification tests ie the Fermat and Euler-Lagrange-Lifchitz tests? If the chain length is only seven then you would only have to perform the tests eight times for each origin or multiple of origin.
 Then you could double the origin and use the verification tests again or you could hash again for a different origin and use the verification tests again.
 Would this just simply be a slower method?
 Or have I misunderstood something?