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Old 2018-10-04, 04:03   #1
miket
 
May 2013

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Default distribution of prime numbers in special range

Let's start from the first prime 2,
primes from 2+1
to 2^2+1 is 3,5,
primes from 5+1
to 5^2+1 is 7,11,13,17,19,23 (6 primes)
primes from 23+1
to 23^2+1 is 29,...523 (90 primes)
primes from 523+1
to 523^2+1 is 541,...273527 (23826 primes)
from wolframalpha, primepi(n^2+1)-primepi(n+1) at n=273527 get 3118310544
the sequence is 1,2,6,90,23826,3118310544
How about the next item of the sequence?
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Old 2018-10-04, 06:32   #2
LaurV
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You ask us to count primes to about 10^20, so the approximate answer is "2,220,819,602,560,918,840", well... hehe... (link)

Last fiddled with by LaurV on 2018-10-04 at 06:32
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Old 2018-10-04, 13:47   #3
Uncwilly
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What is "special" about this range?
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Old 2018-10-04, 14:04   #4
Dr Sardonicus
 
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Quote:
Originally Posted by miket View Post
from wolframalpha, primepi(n^2+1)-primepi(n+1) at n=273527 get 3118310544
the sequence is 1,2,6,90,23826,3118310544
How about the next item of the sequence?
Doesn't wolframalpha tell you? Or does it say something like, "I'm sorry, Dave, I'm afraid I can't do that."?
:-D

Let's see here... The first nine terms of the sequence are

1 primepi(2) - primepi(1)
2 primepi(5) - primepi(2)
3 primepi(23) - primepi(5)
4 primepi(523) - primepi(23)
5 primepi(273527) - primepi(523)
6 primepi(74817019691) - primepi(273527)
7 primepi(5597586435443481735313) - primepi(74817019691)
8 primepi(31332973902260863916713294607866229791207871) - primepi(5597586435443481735313)
9 primepi(981755253559760390192770373112462804615950389612557017724898646851679425786813132352477) - primepi(31332973902260863916713294607866229791207871)

Oh, you want exact numerical values? Well, by term 7 we're beyond most tables of primes, but primepi(5597586435443481735313) might be doable in a not-unreasonable length of time without actually finding all the primes up to 5597586435443481735313.

Alas, I'm too lazy to look up the details.

You can, of course, get estimates on the growth of the sequence of primes, and on the corresponding values of primepi(). The calculation of the next few primes in the sequence may be within reach.

But absent a compelling reason, I'm not going to devote any effort to finding the primepi() values.

Last fiddled with by Dr Sardonicus on 2018-10-04 at 14:06 Reason: fixign tyops
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Old 2018-10-04, 14:22   #5
CRGreathouse
 
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Quote:
Originally Posted by Dr Sardonicus View Post
Oh, you want exact numerical values? Well, by term 7 we're beyond most tables of primes, but primepi(5597586435443481735313) might be doable in a not-unreasonable length of time without actually finding all the primes up to 5597586435443481735313.

Alas, I'm too lazy to look up the details.
It should take less than a CPU-day on
https://github.com/kimwalisch/primecount

A good multi-core CPU might knock it out in an hour.
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