Primality test for numbers of form 10^n + k

Question

I have some numbers of form 10^N + K, where N is about 1000, and K is really small (lower than 500). I want to test these numbers for primality. Currently I'm using Fermat's test by base 2, preceded by checking small factors (<10000).

However, this is rather slow for my purposes. Is there any algorithm quicker than that? Can this special form be exploited somehow?

Also, maybe if two numbers differ only in K, is it possible to test these two numbers a bit quicker?

Answer 1

If K has a factor of either 2 or 5 then 10^N + K is composite. This allows testing a small number quickly. Large primes are all such that P mod 6 = 1 or 5. You can use this to eliminate 2/3 of possible K values. With a little work you can set up a 2-4 wheel to avoid a lot of division:

increment <- either 2 or 4 as required
repeat
  K <- K + increment
  increment <- 6 - increment
  if (K mod 5 = 0) then 
    continue 
  endif
until (isPrime(10^N + K) or (K > 500))

Trial factoring up to 10,000 if fine. Are you building a list of the primes up to 10,000 first? Use Eratosthenes' Sieve to create the list and just read off numbers.

Running a Fermat Test base 2 is a good start, it finds a lot of composites reasonably quickly.

After that you need to implement the probabilistic Miller-Rabin test, and run it enough times that it is more probable your hardware has failed rather than the number is composite.

Answer 2

Also, maybe if two numbers differ only in K, is it possible to test these two numbers a bit quicker?

To check the primes in a relatively small interval like [10^N+K₁, 10^N+K₂] you can use the sieve of Erathostenes to check for divisibility by small numbers.

The remaining prime candidates can then be checked by a probabilistic test like Miller-Rabin.