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Comment by readerrrr | Hacker News Reader

There is such a formula, it is very simple

  n^2+n+41

It doesn't generate sequential primes.

That doesn't really matter, checking if a number is prime can be done in sublinear time complexity. The real problem is factoring large numbers.

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It also stops generating primes for n=40. See

https://en.wikipedia.org/wiki/Formula_for_primes#Prime_formu...

It never stops generating primes, but it doesn't give a prime for every n.

Then why is it better than just

which also doesn't give a prime for any n, but does generate them all.

Because it grows exponentially. ( And it is interesting )

In reality you will use a O( (log n)^6 ) algorithm.

Nobody in their right mind uses AKS for primality testing. You use Miller-Rabin with an appropriate constant number of iterations, with complexity (log n)^(2 + o(1)). Or if you want provability, you use some fast variant of ECPP with (log n)^(4 + o(1)) complexity.

I think n² grows quadratically rather than exponentially.

::blush::

This degrades pretty quickly, however. Only 581 of the first 1000 numbers generated by that formula are prime.

11 also works initially as a constant, but it degrades even more quickly than 41.

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