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MATHEMATICIANS have spent millennia trying to understand prime numbers 鈥� those only divisible by themselves and 1 鈥� but so far no one has discovered a formula for all of them. Now mathematician Simon Plouffe has found a way to produce long sequences of prime numbers, improving on previous efforts.

For example, we have known since the 18th century that n² + n + 41 is prime for values of n up to 39. It breaks down at n = 40, because that gives 1681, which is equal to 41² and thus not prime.

Other types of prime-generating formulae start with a carefully chosen number and use this to generate a string of primes. If you start with n = 1.92878, calculate 2ⁿ and replace n with this new value, you get a sequence of numbers. Ignore everything after the decimal points and the first three numbers in the sequence are prime.

By tweaking the value of n more precisely, you can generate as many primes as you want 鈥� but not all of them. That is because the sequence grows too rapidly, skipping many primes and becoming more difficult to calculate. The function described above generates 3 and then 13, skipping 5, 7 and 11. After that, it jumps to 16,381, followed by a prime 4932 digits long 鈥� that鈥檚 nearly 10⁵⁰⁰⁰ times bigger.

鈥淲e have known since the 18th century that n² + n + 41 is prime for values of n up to 39鈥�

Plouffe has tried to find something that doesn鈥檛 grow too quickly, but will work indefinitely, unlike the n² + n + 41 example. He starts with a carefully chosen prime number, 10⁵⁰⁰ + 961, and uses a formula that generates a string of digits that produce a prime about 100,000 times bigger. Applying the formula again produces another prime, again 100,000 times bigger, so the sequence grows reasonably slowly (arxiv.org/abs/1901.01849).

Using this method, Plouffe was able to generate 50 primes 鈥� more than any other prime-generating algorithm to date.