# Divisibility and PrimesThe Distribution of Primes

The easiest way to check if a number is prime is to try to divide it by all smaller integers. Computers can do this very quickly and efficiently. For *very large* numbers, with hundreds of digits, there are also more efficient algorithms. Some of these even use probability to determine if a number is *almost certainly* prime.

Here is a calculator that allows you to check if any number is prime:

### Prime Checker

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Throughout history, people have tried to find larger and larger prime numbers. In 1460, the largest known prime was 131,071. In 1772,

With the arrival of computers in the 20th century, calculating large primes became much easier. The largest currently known prime was discovered in December 2018 and has 24,862,048 digits. You would need 8000 sheets of paper to print it out!

Calculating these large prime numbers might seem like just a waste of time, but later in this course you’ll learn about various real life applications where computers have to use large primes.

Here you can generate your own prime numbers with a given number of digits:

### Prime Generator

Number of digits:

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## The Ulam Spiral

The Polish mathematician *“long and very boring”* meeting in 1963.

We write down all integers in a rectangular grid, starting with 1 in the middle and then spiralling outwards. Then we highlight all numbers which are prime.

So far, the Ulam spiral doesn’t look particularly exciting. But if we zoom out, interesting patterns emerge. Here are the primes up to 160,000:

Rather than appearing randomly, as one might expect, it seems that certain diagonals are much more popular with primes than others. This creates a curious “plaid” pattern.

*It turns out that these diagonals all correspond to certain quadratic equations which seem to generate prime numbers more often than average. However it is unknown why that would be the case…*

## The Goldbach Conjecture

In 1742, the German mathematician

### Goldbach Calculator

Pick any even number, to calculate how it

can be written as the sum of two primes.

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Goldbach wrote about his observation in a letter to the famous mathematician **Goldbach Conjecture**.

Computers have checked that the Goldbach Conjecture works for every even number up to 4 × 10^{18} (that’s a 4 with 18 zeros), but mathematicians have still not found a proof that it works for *all* even integers. And that is a big difference, because there are infinitely many integers, so we couldn’t possibly check all of them.

Its apparent simplicity made the Goldbach conjecture one of the most famous unsolved problems in mathematics.

## Twin Primes

We have already seen that prime numbers get more spread out as they get bigger. But they always seem to appear completely random, and occasionally we find two primes right next to each other, just one number apart: these are called **Twin Primes**.

35,1113,4143,101103,20272029,108,377108,379,1,523,6511,523,653

The largest known pair of twin primes has an incredible 58,711 digits! But are there infinitely many twin primes, just like there are infinitely many primes? Nobody knows – the *Twin Prime conjecture* is another one of the many unsolved problems surrounding the primes.

## The Riemann Hypothesis

Mathematicians have spent many centuries exploring the pattern and distribution of prime numbers. They seem to appear completely randomly – sometimes there are huge gaps in between consecutive primes, and sometimes we find

When only 15 years old, the German mathematician

Along the x-axis you can see all integers. Whenever there is a prime, the Prime Counting Function (shown in **blue**) increases by one. As we **red**). He predicted that the two functions are always “approximately similar”, and this was proven in 1896.

However, as you can see above, there is still a significant error between the actual number of primes, and Gauss’s approximation. In 1859, the mathematician *always* work. His idea became known as the **Riemann Hypothesis**.

Hundreds of mathematicians have tried to prove Riemann’s hypothesis, but all without success. It is often considered one of the most difficult and most important unsolved problems in mathematics. In 2000, the Clay Mathematics Institute named it one of seven **Millennium Prize Problems**