Frequency of the Prime Numbers
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Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?
For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.
probability number-theory prime-numbers
edited Aug 8 at 10:33
Jam
4,35811330
asked Aug 8 at 10:27
Tsing Shi Tao
444
add a comment |Â
up vote
5
down vote
favorite
Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?
For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.
probability number-theory prime-numbers
edited Aug 8 at 10:33
Jam
4,35811330
asked Aug 8 at 10:27
Tsing Shi Tao
444
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?
For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.
probability number-theory prime-numbers
edited Aug 8 at 10:33
Jam
4,35811330
asked Aug 8 at 10:27
Tsing Shi Tao
444
Suppose I took all natural numbers less than or equal to $x$ and I picked one at random. Is there a way that we know of to express the probability that my number is prime in terms of $x$, for all $x$?
For example, for $x=12$, the prime numbers less than or equal to $x$ are $2,3,5,7$ and $11$, so my probability is $5/12$.
probability number-theory prime-numbers
edited Aug 8 at 10:33
Jam
4,35811330
asked Aug 8 at 10:27
Tsing Shi Tao
444
edited Aug 8 at 10:33
Jam
4,35811330
edited Aug 8 at 10:33
Jam
4,35811330
edited Aug 8 at 10:33
Jam
4,35811330
4,35811330
asked Aug 8 at 10:27
Tsing Shi Tao
444
asked Aug 8 at 10:27
Tsing Shi Tao
444
asked Aug 8 at 10:27
Tsing Shi Tao
444
444
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
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23
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There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.
This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.
answered Aug 8 at 10:29
Toby Mak
2,8051925
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
 |Â
show 4 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
23
down vote
There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.
This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.
answered Aug 8 at 10:29
Toby Mak
2,8051925
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
 |Â
show 4 more comments
up vote
23
down vote
There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.
This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.
answered Aug 8 at 10:29
Toby Mak
2,8051925
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
 |Â
show 4 more comments
up vote
23
down vote
up vote
23
down vote
There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.
This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.
answered Aug 8 at 10:29
Toby Mak
2,8051925
There is no explicit formula, but the prime number theorem says that as $x$ tends to infinity, there are around $fracxln x$ primes $≤ x$.
This means that the chance that a random number is prime will be around $fracxln x cdot frac1x = frac1ln x$.
answered Aug 8 at 10:29
Toby Mak
2,8051925
answered Aug 8 at 10:29
Toby Mak
2,8051925
answered Aug 8 at 10:29
Toby Mak
2,8051925
answered Aug 8 at 10:29
Toby Mak
2,8051925
2,8051925
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
 |Â
show 4 more comments
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
4
4
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
In OPs example, 1/ln(12) = 0.40 while 5/12 = 0.42.
– infinitezero
Aug 8 at 12:34
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
infinity is a bit further
– dEmigOd
Aug 8 at 12:35
1
1
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
@infinitezero The formula is not exact – I said the chance will be around that number. (By the way, this answer is blowing up!)
– Toby Mak
Aug 8 at 12:36
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
I know. I just pointed out that even for small X it's pretty close.
– infinitezero
Aug 8 at 12:37
5
5
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
$frac1ln x$ is the chance that a random number selected from the set $1,...,x$ will be prime.
– probably_someone
Aug 8 at 13:22
 |Â
show 4 more comments
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