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January 20th, 2014, 08:08 PM   #1
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primes and twin primes: Number between powers of 10

I have been curious about prime and twin prime distribution. So now lets explore primes and twin primes.

1-100: Primes:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61, 67,71,73,79,83,87,91,97

1-100: Twin Primes

3,5,7,11,13,17,19,41,43,71,73

100-1000: Primes:

101,103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167,173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239,241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467,479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733,739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823,827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911,919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

100-1000: Twin Primes

101,103,107,109137,139,191,193,197,199,227,229,281 ,283,311,313,347,349431,433,461,463,521,523,641,64 3,827,829,857,859,881,883



Now why is it that as you go from 1 power of ten to another power of ten the number of primes increases but the percent primes decreases? now do twin primes and primes have a constant ratio or does the ratio change?
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January 20th, 2014, 11:32 PM   #2
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Re: primes and twin primes: Number between powers of 10

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now do twin primes and primes have a constant ratio or does the ratio change?
Asymptotically 0. Number of twin primes upto some large n is c * n/log(n)^2 for some c (Brun sieve) whereas primes are n/log(n), so ratio is c/log(n).
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January 21st, 2014, 09:03 AM   #3
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Re: primes and twin primes: Number between powers of 10

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Originally Posted by caters
now do twin primes and primes have a constant ratio or does the ratio change?
Asymptotically 0. Number of twin primes upto some large n is c * n/log(n)^2 for some c (Brun sieve) whereas primes are n/log(n), so ratio is c/log(n).


thats only approximate not exact like you would get if you counted all the primes and all the twin primes and divided the twin primes by the primes(the actual number of them not the primes themselves).
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January 21st, 2014, 09:16 AM   #4
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Re: primes and twin primes: Number between powers of 10

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thats only approximate not exact like you would get if you counted all the primes and all the twin primes and divided the twin primes by the primes(the actual number of them not the primes themselves).
You asked if the ratio was constant or not, and I think this was the answer you were looking for. To recap: the ratio is always positive (as long as you're looking at least up to 10^1), but it tends toward 0. For example, it's only above 1/googol finitely often.
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January 21st, 2014, 11:48 AM   #5
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Re: primes and twin primes: Number between powers of 10

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thats only approximate not exact like you would get if you counted all the primes and all the twin primes and divided the twin primes by the primes(the actual number of them not the primes themselves).
It's not an approximation, but a precise bound. You would get asymptotically correct errors and it grows large only finitely often.
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January 21st, 2014, 08:50 PM   #6
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Re: primes and twin primes: Number between powers of 10

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Originally Posted by caters
thats only approximate not exact like you would get if you counted all the primes and all the twin primes and divided the twin primes by the primes(the actual number of them not the primes themselves).
It's not an approximation, but a precise bound. You would get asymptotically correct errors and it grows large only finitely often.

counting the primes and twin primes and dividing the number of twin primes by the total primes like this:

# of twin primes/ # of primes

gives you an exact ratio. It might give you repeating decimals but that is still exact.

c/log(n) (I think that is what you said) gets closer and closer to the exact ratio but still doesn't get to it exactly.

thus you have this:

lim x(where x is c/log(n)) -> ? = # of twin primes/ # of primes

It is an approximate that equation you have. Simply counting the number of primes and twin primes gives you the exact ratio.
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January 22nd, 2014, 01:07 AM   #7
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Re: primes and twin primes: Number between powers of 10

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Simply counting the number of primes and twin primes gives you the exact ratio.
You cannot "simply count" prime numbers. Period.
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January 22nd, 2014, 05:20 AM   #8
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Re: primes and twin primes: Number between powers of 10

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Originally Posted by caters
Simply counting the number of primes and twin primes gives you the exact ratio.
You cannot "simply count" prime numbers. Period.

by simply counting I don't mean just looking at the numbers and add 1 to 1 every time you come across a primes(that would mean much fewer primes than ?). I mean doing trial division and figuring out it is prime and than adding 1 to the number of primes you have so far.
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January 22nd, 2014, 06:25 AM   #9
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Re: primes and twin primes: Number between powers of 10

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by simply counting I don't mean just looking at the numbers and add 1 to 1 every time you come across a primes(that would mean much fewer primes than ?). I mean doing trial division and figuring out it is prime and than adding 1 to the number of primes you have so far.
A far powerful formula is to count all the primes by using Riemann's formula but that would give asymptotically the same conclusion as mine : The ratio tends to 0.
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January 22nd, 2014, 06:40 AM   #10
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Re: primes and twin primes: Number between powers of 10

Quote:
Originally Posted by caters
counting the primes and twin primes and dividing the number of twin primes by the total primes like this:

# of twin primes/ # of primes

gives you an exact ratio.
Counting up to what bound?
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