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April 1st, 2019, 04:11 AM  #1 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12  number of primes less than n
is this correct? pi(n) = number of primes less than n pi(n) = n × (1/2 × 2/3 × 4/5 × ... × p1/p) p = largest prime less than square root of n 
April 1st, 2019, 04:52 AM  #2 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
proof a prime number is a number that isn't divisible by primes less than or equal to it's square root 1/2 of numbers are not divisible by 2 2/3 of them are not divisible by 3 4/5 of them are not divisible by 5 . . . p1/p of them are not divisible by p p = largest prime equal to or less than square root of given number so number of primes less than n is n × product of these fractions 
April 1st, 2019, 07:29 AM  #3 
Senior Member Joined: Aug 2012 Posts: 2,321 Thanks: 714 
Half of the numbers aren't divisible by 2. Then 2/3 of the numbers aren't divisible by 3 ... but you counted 6 twice so you have to account for that. Likewise 4/5 aren't divisible by 5 but you counted 10 and 15 twice, etc. So your formula works once you put in all the correction factors, which make it a lot more complicated.
Last edited by Maschke; April 1st, 2019 at 07:54 AM. 
April 1st, 2019, 08:55 AM  #4 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
I didn't count composite numbers like 6 or 10. I counted prime numbers; half of numbers are not divisible by 2 (odd numbers) 1 3 5 7 9 11 as you see, 2/3 of them (odd numbers) are not divisible by 3 (1 5 7 11) 1/2 × 2/3 = 1/3 of numbers are not divisible by 2 or 3 Last edited by skipjack; April 1st, 2019 at 10:13 AM. 
April 1st, 2019, 09:37 AM  #5  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,194 Thanks: 897 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49 for example. I count 15 primes out of 25 numbers. The ratio is closer to 1/2 than 1/3. This shows that your ratio of 2/3 is only an approximation. Part of the trouble here is that you have to have the whole list of primes to do your ratios. Unfortunately, that list is infinite, so you are stuck with approximations. Dan Last edited by skipjack; April 1st, 2019 at 10:14 AM.  
April 1st, 2019, 10:25 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Of those 25 numbers, 14 are prime and 10 are composite. The 10 composite numbers comprise 7 that divide by 3, 2 that divide by 5 (but not by 3) and 1 that divides by 7 (but not by 3 or 5). Hence youngmath's method works quite well for the above example. 
April 1st, 2019, 10:30 AM  #7 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
this formula does't count prime numbers less than square root of n true formula is: pi(n)  pi(√n) = n × (1/2 × 2/3 × 4/5 × ... × p1/p) p = largest prime less than or equal to square root of n 
April 1st, 2019, 10:42 AM  #8 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
pi(10000) = 10000 × (1/2 × 2/3 × ... × 96/97) + pi(100) = 1203 + 25 = 1228 and true value of pi(10000) is 1229

April 1st, 2019, 10:52 AM  #9  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,194 Thanks: 897 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  

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