Prime number theorem facts for kids

The prime number theorem is a special rule in number theory. It helps us understand how prime numbers are spread out among all the other numbers.

Prime numbers don't show up in a regular pattern. They seem to appear randomly. But this theorem helps us see a general trend. It tells us that as numbers get bigger, it becomes less likely to find a prime number.

What is the Prime Number Theorem?

The prime number theorem explains that the chance of finding a prime number gets smaller as you look at larger numbers. Imagine you're picking a random number. If you pick from numbers up to 100, you have a certain chance of finding a prime. If you pick from numbers up to 1,000,000, your chance of finding a prime is much lower.

How Does It Work?

The theorem uses a special math idea called the natural logarithm. Don't worry, it's not too complicated! The natural logarithm of a number helps us estimate how many digits that number has, or how big it is in a special way.

The theorem says that the chance of a number 'n' being prime is roughly 1 divided by its natural logarithm (1/ln(n)).

Prime Numbers and Digits

This means something cool about numbers with many digits. For example, if a number has twice as many digits as another, it's about half as likely to be a prime number.

Let's look at an example:

• Among numbers with up to 1000 digits, about 1 in 2300 is prime.
• Among numbers with up to 2000 digits, about 1 in 4600 is prime.

See how the chance gets smaller? This also means that the average space between prime numbers gets bigger as the numbers themselves get bigger.

Who Discovered It?

The idea behind the prime number theorem was first thought of by a very smart mathematician named Carl Friedrich Gauss. He was only 15 years old in 1793 when he suspected a link between prime numbers and logarithms.

Another mathematician, Adrien-Marie Legendre, also had similar ideas in 1798.

It took over a century for the theorem to be fully proven! Two mathematicians, Jacques Hadamard and Charles-Jean de La Vallée Poussin, finally proved the prime number theorem in 1896.

See also

In Spanish: Teorema de los números primos para niños