In primes_upto, use the Sieve.

Using the Sieve properly, division is not used, only addition (and the odd multiplication). Make a list of odd numbers up to the target. (Magically include 2 as the first prime, but it doesn't need to be in the list.) With 3 as the first odd prime, mark off all multiples of 3 up to the target. Search from 3 upwards until an unmarked number is found (5) - this is the next prime. Mark all multiples of this as composite. Lather, rinse, repeat.

One optimization is to realize that all numbers less than the square of the next prime have been marked correctly as prime or composite. For instance, marking with 3, 5 is next, and all numbers less than 25 are correctly marked. When marking with 5, you can start marking with 25, since it has no factors less than 5. Any composite between 5 and 25 will already be marked, since it will have at least 1 factor less than 5. This quickly leapfrogs over previously covered ground. Since the primes also get bigger, the jump between marks does as well.

Another is to realize that once all primes less than or equal to the square root of the target are found, marking stops. Any unmarked numbers above the square root are prime.

-QM
--
Quantum Mechanics: The dreams stuff is made of

In reply to Re: Math help: Finding Prime Numbers by QM
in thread Math help: Finding Prime Numbers by Ovid

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