How do I generate Primes Using 6*k +- 1 rule

Question

We know that all primes above 3 can be generated using:

6 * k + 1
6 * k - 1

However we all numbers generated from the above formulas are not pr

Answer 1

You don't need to add all possible candidates to the array. You can create a Set to store all non primes.

Also you can start checking at k * k, rather than 2 * k

  public void primesTo1000() {
    Set notPrimes = new HashSet<>();
    ArrayList primes = new ArrayList<>();
    primes.add(2);//explicitly add
    primes.add(3);//2 and 3

    for (int i = 1; i < (1000 / 6); i++) {
      handlePossiblePrime(6 * i - 1, primes, notPrimes);
      handlePossiblePrime(6 * i + 1, primes, notPrimes);
    }
    System.out.println(primes);
  }

  public void handlePossiblePrime(
      int k, List primes, Set notPrimes) {
    if (!notPrimes.contains(k)) {
      primes.add(k);
      for (int j = k * k; j <= 1000; j += k) {
        notPrimes.add(j);
      }
    }
  }

untested code, check corners

---

Here is a bit packing version of the sieve as suggested in the answer referenced by @Will Ness. Rather than return the n_th prime, this version returns a list of primes to n:

public List primesTo(int n) {
  List primes = new ArrayList<>();
  if (n > 1) {
    int limit = (n - 3) >> 1;
    int[] sieve = new int[(limit >> 5) + 1];
    for (int i = 0; i <= (int) (Math.sqrt(n) - 3) >> 1; i++)
      if ((sieve[i >> 5] & (1 << (i & 31))) == 0) {
        int p = i + i + 3;
        for (int j = (p * p - 3) >> 1; j <= limit; j += p)
          sieve[j >> 5] |= 1 << (j & 31);
      }
    primes.add(2);
    for (int i = 0; i <= limit; i++)
      if ((sieve[i >> 5] & (1 << (i & 31))) == 0)
        primes.add(i + i + 3);
  }
  return primes;
}

---

There seems to be a bug in your updated code that uses a boolean array (it is not returning all the primes).

public static List booleanSieve(int n) {
  boolean[] primes = new boolean[n + 5];
  for (int i = 0; i <= n; i++)
    primes[i] = false;
  primes[2] = primes[3] = true;
  for (int i = 1; i <= n / 6; i++) {
    int prod6k = 6 * i;
    primes[prod6k + 1] = true;
    primes[prod6k - 1] = true;
  }
  for (int i = 0; i <= n; i++) {
    if (primes[i]) {
      int k = i;
      for (int j = k * k; j <= n && j > 0; j += k) {
        primes[j] = false;
      }
    }
  }

  List primesList = new ArrayList<>();
  for (int i = 0; i <= n; i++)
    if (primes[i])
      primesList.add(i);

  return primesList;
}

public static List bitPacking(int n) {
  List primes = new ArrayList<>();
  if (n > 1) {
    int limit = (n - 3) >> 1;
    int[] sieve = new int[(limit >> 5) + 1];
    for (int i = 0; i <= (int) (Math.sqrt(n) - 3) >> 1; i++)
      if ((sieve[i >> 5] & (1 << (i & 31))) == 0) {
        int p = i + i + 3;
        for (int j = (p * p - 3) >> 1; j <= limit; j += p)
          sieve[j >> 5] |= 1 << (j & 31);
      }
    primes.add(2);
    for (int i = 0; i <= limit; i++)
      if ((sieve[i >> 5] & (1 << (i & 31))) == 0)
        primes.add(i + i + 3);
  }
  return primes;
}

public static void main(String... args) {
  Executor executor = Executors.newSingleThreadExecutor();
  executor.execute(() -> {
    for (int i = 0; i < 10; i++) {
      int n = (int) Math.pow(10, i);
      Stopwatch timer = Stopwatch.createUnstarted();
      timer.start();
      List result = booleanSieve(n);
      timer.stop();
      System.out.println(result.size() + "\tBoolean: " + timer);
    }

    for (int i = 0; i < 10; i++) {
      int n = (int) Math.pow(10, i);
      Stopwatch timer = Stopwatch.createUnstarted();
      timer.start();
      List result = bitPacking(n);
      timer.stop();
      System.out.println(result.size() + "\tBitPacking: " + timer);
    }
  });
}

---

0   Boolean: 38.51 μs
4   Boolean: 45.77 μs
25  Boolean: 31.56 μs
168 Boolean: 227.1 μs
1229    Boolean: 1.395 ms
9592    Boolean: 4.289 ms
78491   Boolean: 25.96 ms
664116  Boolean: 133.5 ms
5717622 Boolean: 3.216 s
46707218    Boolean: 32.18 s
0   BitPacking: 117.0 μs
4   BitPacking: 11.25 μs
25  BitPacking: 11.53 μs
168 BitPacking: 70.03 μs
1229    BitPacking: 471.8 μs
9592    BitPacking: 3.701 ms
78498   BitPacking: 9.651 ms
664579  BitPacking: 43.43 ms
5761455 BitPacking: 1.483 s
50847534    BitPacking: 17.71 s