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Old 2017-04-30, 15:31   #23
Batalov
 
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Quote:
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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.
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Old 2017-04-30, 16:59   #24
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Quote:
Originally Posted by Batalov View Post
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!
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Old 2022-07-03, 01:40   #25
Batalov
 
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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
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Old 2022-07-13, 19:33   #26
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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
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Old 2022-07-14, 01:14   #27
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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?!
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Old 2022-07-14, 06:57   #28
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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
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Old 2022-07-14, 07:11   #29
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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, <next term?> - 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)
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Old 2022-07-14, 07:32   #30
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Quote:
Originally Posted by Batalov View Post
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, <next term?>
This is really interesting, for the case of Perrin sequence, Perrin(355) and Perrin(356) are both primes, and next term is > 10^6
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Old 2022-07-15, 00:20   #31
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Quote:
Originally Posted by Batalov View Post
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.
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Old 2022-07-15, 05:07   #32
Batalov
 
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Quote:
Originally Posted by Batalov View Post
- 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
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Old 2022-07-15, 08:27   #33
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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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