There are 25 primes between 0 and 100 but only 168 between 0 and 1000 where 250 may be expected. Why thinning?

Question

Why are the number of primes not distributed evenly?

Answer 1

The number of primes <=x goes as x/log(x) (this is known as the Prime Number Theorem), so there are fewer primes in an interval of given length as you go farther out.

Answer 2

The density of primes is not linear. Legendre and Gauss worked on this and came to the conclusion that it is proportional to 1/log(n) for large n.

Let 's see here 1/log(100) = 1/2 which relates to 25/100
1/log(1000) = 1/3 which relates to 168/1000

In other words,
168/1000 = d1 * 1/3
25/100 = d2 * 1/2

d1 = 0.504
d2 = 0.5

As you can see d1 is very close to d2 so it seems the proportion is holding.

Answer 3

I guess I am unsure why you expected 250 primes between 0 and 1000.

There is a function (called the pi function) which is the number of primes less than x. pi(x) is closely approximated by x/(log(x - 1)) *when* x gets very large. For a complete explanation with illustrations you can check this out:

http://primes.utm.edu/howmany.shtml

HTH

Charles

Answer 4

Why should they be? There is no reason to expect that. The larger the number range, a relatively greater number of composites become possible.

Think about an analogous question:

"There are 2(3) perfect "squares" in the range (1, 10) but only 9(10) in the range (1, 100), depending on whether you agree to include 1 or not. Why aren't there the 'expected' 20(30) in the range (1, 100)? Why thinning?"

It is naive to think that any given property of numbers should simply scale linearly. The distribution of primes among the integers is in fact a very significant problem in Number Theory, connected to many other important topics. The simplest statement is the prime number theorem: the PROPORTION of primes less than x is asymptotic to 1/ln x. (On closer examination, mathematicians have determined a constant term and other variable factors modifying this simplest asymptotic statement.)

Live long and prosper.

Answer 5

Take two random numbers, say x and y, where y is 100 times larger than x.

To be a prime, x must not be a multiple of any prime number that is below x.

To be a prime, y must not be a multiple of any prime number that is below y. That's a lot more conditions than x has to meet.

Answer 6

larger numbers have fewer primes.

that's b/c more of them are products of smaller numbers.

the larger the number, the more numbers are smaller than it, the higher the chance that it will be a product of some of them.