Prime number generator, how to make Can anybody point me an algorithm to generate prime numbers, I know of a few ones (Mersenne, Euclides, etc.) but they fail to generate much primes...
The objective is: 
    given a first prime, generate the 'n' next primes. But thanks for the link ;-)
for example : primes( 17, 50 ) -> Generate 50 consecutive primes starting at 17

and do not fail any prime in this 50... no holes!
 A: "The objective is: given a first prime, generate the $n$ next primes." 
This is equivalent to, "given an integer $N$, find the smallest prime exceeding $N$." 
People do this all the time, for example, here is a tabulation of the smallest prime exceeding $10^m$ for various values of $m$. But there's no especially clever way to do it. In effect, you look at $N+1,N+2,N+3,\dots$ until you find a prime. You can save some work by sieving, but you already know that. If you want to find, say, the first 50 primes after $10^{100}$, no doubt you'd save a lot of work by doing some sieving, but in the end there would be a lot of numbers that get through the sieve, and you'd have to fall back on the standard primality tests to see which ones are actually prime. 
I guess the other thing you can do is use the Prime Number Theorem to estimate how far you have to sieve to have a good chance of catching the next 50 primes. Roughly speaking, one number in every $\log N$ will be prime, so if you go out to $N+100\log N$ you should have an excellent chance of catching the next 50 primes. 
A: Here is a paper that contains a prime recurrence defined by
$$a_{n+1} = a_n + gcd(n,a_{n-1}),$$
where $gcd$ is the greatest common divisor function.
A favorite of mine is given by Mills' theorem, but since we cannot compute the number directly (yet...?), it is not feasible to generate primes with it.
A: The fundamental theorem of arithmetic as a recurrence gives all the primes. But of course, it involves a lot of 1:s which are not primes, and also it is a 2-dimensional matrix.
The recurrence in European dot-comma English Excel to be entered in cell A1, is:
=IF(COLUMN()=1;1;IF(ROW()=COLUMN();ROW()/PRODUCT(INDIRECT(ADDRESS(ROW();1)
&":"&ADDRESS(ROW();COLUMN()-1)));IF(ROW()>COLUMN();INDIRECT(ADDRESS(ROW()
-COLUMN();COLUMN()));"")))

which outputs:

where the sequence in the diagonal is the exponentiated von Mangoldt function.

Edit 29.3.2013:
A more Riemann zeta function like table can be done with the recurrence:
=IF(ROW()>=COLUMN();IF(AND(ROW()=1;COLUMN()=1);1;IF(COLUMN()=1;
ROW()/PRODUCT(INDIRECT(ADDRESS(ROW();2)&":"&ADDRESS(ROW();ROW())));
IF(MOD(ROW();COLUMN())=0;INDIRECT(ADDRESS(FLOOR(ROW()/COLUMN();1);
1));"")));"")

which outputs:

where again the exponentiated von Mangoldt function is found in the first column.
However this second recurrence uses the floor function.
A: When you mentioned prime numbers, I thought you would like quite large prime numbers. Thinking from applications in computer science, I thought cryptography would give an answer, especially RSA that uses large prime numbers. 
Indeed, the crypto.SE had the following question that I believe solves your problem: https://crypto.stackexchange.com/questions/71/how-can-i-generate-large-prime-numbers-for-rsa
Edit:
As you mentioned, you are interested in generating the n smallest primes that are greatest than a certain number. There are some ways to do that, although I cannot judge how efficient they are computationally (some require knowing all primes that are smaller than the one you are looking for). I suggest consulting the excellent article on wikipedia http://en.wikipedia.org/wiki/Formula_for_primes , as well as the related links and references.
A: Here you go - give any random number as an input, and it will give you the next prime after it. You can feed its output into itself to make a list. It basically uses Fermat's Little Theorem to heuristically check if each number is a prime, and keeps checking successive odd numbers until you get to something that works.
It has a very small chance of failure (eg. nextprime(560) = 561, but 561=3*11*17), but if you go high enough this becomes negligible in practice.
def modexp(b,e,n):
  if e == 0: return 1
  elif e%2 == 0: return modexp(b,e/2,n)**2 % n
  else: return b*modexp(b,e/2,n)**2 % n

def nextprime(inp):
  inp += inp%2 + 1
  while modexp(2,inp,inp) != 2 or modexp(3,inp,inp) != 3: inp += 2
  return inp

A: Looking at this from a computer science like perspective, I would use the following algorithm to generate my list of primes. 
input = (17,50)
    if 17 is prime: 
    counter = 1
    from_seed = seed
      while counter <= 50
        if from_seed is prime
         print from_seed
         counter+1
         from_seed+1
        else
       from_seed+1

To check if the number is prime then I would do the very simple
boolean isPrime (n)
   if n <= 2
     return false
   else
     for i .. sqrt(n)
       if n % i == 0
         return false 
   return true

This will very easily generate what you want. Below I have the working Java code, if you wish to implement it. 
public class PrimeGeneration_Seed {

    public static void main(String[] args) {
        prime_Generation(17,50);
    }

    public static void prime_Generation (int seed, int number_of_primes) {
        if(!isPrime(seed)) {
            System.out.println("Seed is not prime"); // simple out statement saying that "seed" is not prime
        }else {
            int primes = 1;
            int from_seed = seed;
            while(primes <= number_of_primes) {
                if(isPrime(from_seed)) {
                    System.out.println(from_seed);
                    primes++;
                    from_seed++;
                }else {
                    from_seed++;
                }
            }
        }

    }

    // simple primality test - can be modified to be more complex and more efficient
    public static boolean isPrime(int n) {
        if(n <= 2){
            return false;
        }else {
            for(int i = 2;i<=Math.sqrt(n);i++) {
                if(n % i == 0) {
                    return false;
                }
            }
        }
        return true;
    }

}

You can access this link to see the program in action.
