The prime numbers are those integers that can only be divided by themselves and 1. The first seven are
Every other positive integer can be written as a product of prime numbers in a unique way — for example — so the prime numbers are like basic building blocks that other integers can be constructed from. This is why people find them interesting.
We have known for thousands of years that there are infinitely many prime numbers (see here for a proof), but there isn't a simple formula which tells us what they all are. Powerful computer algorithms have enabled us to find larger and larger primes, but we will never be able to write down all of them.
The prime number theorem tells us something about how the prime numbers are distributed among the other integers. It attempts to answer the question "given a positive integer , how many integers up to and including are prime numbers"?
The prime number theorem doesn’t answer this question precisely, but instead gives an approximation. Loosely speaking, it says that for large integers , the expression
is a good estimate for the number of primes up to and including , and that the estimate gets better as gets larger. Here is the natural logarithm of , which you can find on your calculator.
As an example, let's take . The actual number of primes up to and including (which you can look up in this list) is . Our estimate is
which is a pretty good approximation.
However, to be precise about what the prime number theorem tells us, we need to say what we mean by "a good estimate". The prime number theorem does not say that for large values of the difference between the true value and our approximation is close to Instead, it tells us something about the question "what’s the approximation as a percentage of the true value"?
To go back to our example of , the true value was and the approximation was . Therefore, the approximation constitutes a proportion of
of the true result, which translates to 86%. Not bad.
For the actual number of primes up to and including is , so that’s the true value, and the estimate was
Therefore in this case the estimate constitutes a proportion of
of the true value. This translates to 90%, so here the estimate is better than for .
Generally, the prime number theorem tells us that for large the approximation is nearly 100% of the true value. In fact you can get it to be as close to 100% as you like by choosing a large enough
The red curve shows the number of primes up to and including n, where n is measured on the horizontal axis. The blue curve gives the value of n/ln(n). The difference between true result and approximation increases as n grows, but the ratio between the two values tends to 1.
To state the prime number theorem in its full mathematical glory using mathematical notation, let’s first write for the number of primes that are smaller than, or equal to, . The theorem says that
If you know a bit of calculus you will know that you can swap the numerator and denominator here, so expression (1) is equivalent to
The prime number theorem is usually stated using the second expression (2). It is also sometimes written as
which in spoken language is " is asymptotic to as tends to infinity."
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