[seqfan] yet more primes
Chuck Seggelin
barkeep at plastereddragon.com
Fri Dec 26 05:46:39 CET 2003
> Others have sought prime lists, wherein one must always add the offset.
> (So there's no difference between population and depth.) There are two
> kinds of particular interest.
>
> 1) Cunningham chains:
> The multiplier is two, and the offset is either +1 or -1.
> So far, they've found:
> 2n-1, starting at 69257563144280941; length is 15 (see A005603,
A064812)
> 2n+1, starting at 95405042230542329; length is 14 (see A005602)
> When converted to trees (offset +-1), those starting points yield no new
> primes; but they're still deeper than (50n+-27, from 1879).
Thanks for pointing this out to me! In the intervening days I've found 3
prime trees 15 generations deep:
2p+-15 from 13
2p+-4095 from 13043
2p+-4095 from 2999
And 7 prime trees 14 generations deep:
2p+-15 from 11
2p+-255 from 19219
2p+-4095 from 17093
2p+-4095 from 10093
2p+-4095 from 17659
2p+-4095 from 30181
4p+-63 from 47
I'm convinced that eventually I should be able to find a prime tree of depth
16, making it deeper than the record holder Cunningham Chain.
> 2) Arithmetic sequences of primes
> Here the multiplier is one. A record-holder is
> 1n+4609098694200, starting at 11410337850553; length is 22
> (see http://mathworld.wolfram.com/PrimeArithmeticProgression.html )
> But it's silly to make a tree with a multiplier of one.
It seems as though there ought to be a prime tree out there somewhere deeper
than 22 generations. Since every node in a prime tree can have up to two
children, whereas each "node" in a Cunningham Chain or arithmetic
progression can have only one child, it appears that the odds are in the
favor of prime trees for greater depth. (Though I have no idea where to
start looking for trees of such depth.)
I've also found some remarkably wide prime trees in the last couple days.
The previous record was 6p+-5 from 2 which had a population of 28 nodes.
2p+-15 from 13 has a burgeoning population of 71 nodes! (I have
affectionately nicknamed 2p+-15 from 13 "the monster" for this reason.)
Other "fat" trees discovered recently are:
56 nodes: 2p+-15 from 11
46 nodes: 4p+-4095 from 1237
38 nodes: 3p+-80 from 37
38 nodes: 6p+-35 from 179
30 nodes: 5p+-18 from 5
I have concluded that it was a mistake to hold off exploring trees where the
offset (d) exceeded the coefficient (c). I determined that my reasoning
that such cases were duplicative was erroneous. I have found that trees
where d = c raised to some power minus 1 tend to be very generative.
Happy Holidays everyone!
-- Chuck Seggelin
PS: Here's the 15-generation, 71-node "monster" (2p+-15 from 13) for the
curious:
+-----------------------------------------------------------------
|[13]
| - = [11]
| . - = [7]
| . . + = [29]
| . . - = [43]
| . . . - = [71]
| . . . . - = [127]
| . . . . . - = [239]
| . . . . . . - = [463]
| . . . . . . - = [911]
| . . . . . . + = [941]
| . . . . . . - = [1867]
| . . . . . . - = [3719]
| . . . . . + = [269]
| . . . . . - = [523]
| . . . . . - = [1031]
| . . . . . + = [1061]
| . . . . . + = [2137]
| . . . . . - = [4259]
| . . . . . + = [4289]
| . . . . . - = [8563]
| . . . . + = [157]
| . . . + = [101]
| . . + = [73]
| . . - = [131]
| . . + = [277]
| . . + = [569]
| . . - = [1123]
| . . + = [1153]
| . + = [37]
| . - = [59]
| . . - = [103]
| . . - = [191]
| . . - = [367]
| . . . - = [719]
| . . . - = [1423]
| . . . . + = [2861]
| . . . . + = [5737]
| . . . . + = [11489]
| . . . . - = [22963]
| . . . . + = [22993]
| . . . . - = [45971]
| . . . . + = [91957]
| . . . + = [1453]
| . . + = [397]
| . . + = [809]
| . + = [89]
| . - = [163]
| . . - = [311]
| . . - = [607]
| . . + = [1229]
| . . + = [2473]
| . . - = [4931]
| . + = [193]
| . + = [401]
| . - = [787]
| . - = [1559]
| + = [41]
| - = [67]
| . + = [149]
| . - = [283]
| . + = [313]
| . + = [641]
| . + = [1297]
| . - = [2579]
| . + = [2609]
| . + = [5233]
| + = [97]
| - = [179]
| + = [373]
| + = [761]
+-----------------------------------------------------------------
Ordered list of nodes in 2p+-15 from 13:
7, 11, 13, 29, 37, 41, 43, 59, 67, 71, 73, 89, 97, 101, 103, 127, 131, 149,
157, 163, 179, 191, 193, 239, 269, 277, 283, 311, 313, 367, 373, 397, 401,
463, 523, 569, 607, 641, 719, 761, 787, 809, 911, 941, 1031, 1061, 1123,
1153, 1229, 1297, 1423, 1453, 1559, 1867, 2137, 2473, 2579, 2609, 2861,
3719, 4259, 4289, 4931, 5233, 5737, 8563, 11489, 22963, 22993, 45971, 91957
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