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2017-04-30, 15:31   #23
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

9,901 Posts

Quote:
 Originally Posted by Batalov Partition function is available in GP/Pari (p(n) = numbpart(n)), but it is slow for large values of n. It is possible to calculate p(n) with the Arb implementation. ...the first (PR)prime value for n>=1010 is p(10000076282) and has 111391 decimal digits.
To put a symmetrical coda on this thread:
the first (PR)prime value for n>=1011 is p(100000135540) and has 352269 decimal digits.

2017-04-30, 16:59   #24
paulunderwood

Sep 2002
Database er0rr

22·1,063 Posts

Quote:
 Originally Posted by Batalov To put a symmetrical coda on this thread: the first (PR)prime value for n>=1011 is p(100000135540) and has 352269 decimal digits.
Congrats!

 2022-07-03, 01:40 #25 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 100110101011012 Posts Recently some folks submitted a few more prime partition numbers to UTM Partitions primes. I ran some step-stone-"just test that CM-ecpp works" candidates to prove and an interesting factoid reared its (fairly trivial) head: a) generally, on average there could be a prime partitions number for each decimal length, and ... b) there is a prime with exactly 20,000 decimal digits in length, but: c) there isn't with 25,000 decimal digits (smallest above is 25,002 decimal digits long) and d) there isn't with 30,000 decimal digits (smallest above is 30,001 decimal digits long)... e) there is a prime with exactly 40,000 decimal digits in length
 2022-07-13, 19:33 #26 paulunderwood     Sep 2002 Database er0rr 109C16 Posts After almost filling the top20 Partitions primes table with numbers above 13k digits, with our E6 prover code, and starting to eat our own tail, we notice Serge's submissions at 14k digits and so we are now embarking on index 350000000+ i.e. over 20k digits. Last fiddled with by paulunderwood on 2022-07-13 at 19:45
 2022-07-14, 01:14 #27 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 100110101011012 Posts I decided to run some that will push the 13k-ers out of relevancy by the top-20 limit. Or else Chuck likely has another few hundred to run ... one after another, day by day. My imagination is too limited to understand the point of proving each consecutive 13,000+ digit prime partition number. Maybe if I do 15 of the same (but 14,000+ digits) I will get enlightened and will understand?!
 2022-07-14, 06:57 #28 paulunderwood     Sep 2002 Database er0rr 22·1,063 Posts You'll be enlightened soon. CM is is a joy with its MPI abilities. Like you say, a prime a day. It is so much fun in comparison to running Primo. Chuck is cutting back to one every 10 days
 2022-07-14, 07:11 #29 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 9,901 Posts Same random faceless numbers. Just a bit larger. Wouldn't it be more interesting to find them with a bit of spice, like - p(n) and p(n+1) are prime. E.g. n = 2, 1085, - A355728 - p(n) is prime and a member of a twin prime pair - A355704, A355705, A355706 - p(n2) is prime. Oh. Wait. I've already done this sequence (I did cubes too)
2022-07-14, 07:32   #30
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

7×509 Posts

Quote:
 Originally Posted by Batalov Wouldn't it be more interesting to find them with a bit of spice, like - p(n) and p(n+1) are prime. E.g. n = 2, 1085,
This is really interesting, for the case of Perrin sequence, Perrin(355) and Perrin(356) are both primes, and next term is > 10^6

2022-07-15, 00:20   #31
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

20728 Posts

Quote:
 Originally Posted by Batalov Same random faceless numbers. Just a bit larger.
Prime hunting never seemed to me to be so extremely addictive: so many stages0 where somebody can get caught in!

0: Meta level.

2022-07-15, 05:07   #32
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

9,901 Posts

Quote:
 Originally Posted by Batalov - p(n) is prime and a member of a twin prime pair
Wrote a sieve and ran some range and found a few large sequence elements (e.g. p(2335166)); will run for some more.
And credit where credit is due: look at the state of p(2335166) in FactorDb, eh? It has been proven by Greg around March, so Greg also is not a foreigner in the land of fun. (Interestingly p(2335166) existed, but p(2335166)+2 didn't exist in FactorDb until I entered it.)

A355704, A355705, A355706

2022-07-15, 08:27   #33
charybdis

Apr 2020

22×199 Posts

Quote:
 Originally Posted by sweety439 Also, I conjectured that for all integer k>=1, there are only finitely many n such that p(n) and p(n+k) are both primes
Ah, yet another Sweety conjecture that's almost certainly false...

Hint: do the partition numbers grow exponentially?
Second hint: the probability that p(n) is prime is of order 1/sqrt(n)

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