# Vantieghems Theorem for Primality Test

Vantieghems Theorem is a necessary and sufficient condition for a number to be prime. It states that for a natural number n to be prime, the product of where , is congruent to .

In other words, a number n is prime if and only if.

Examples:

• For n = 3, final product is (21 – 1) * (22 – 1) = 1*3 = 3. 3 is congruent to 3 mod 7. We get 3 mod 7 from expression 3 * (mod (23 – 1)), therefore 3 is prime.
• For n = 5, final product is 1*3*7*15 = 315. 315 is congruent to 5(mod 31), therefore 5 is prime.
• For n = 7, final product is 1*3*7*15*31*63 = 615195. 615195 is congruent to 7(mod 127), therefore 7 is prime.
• For n = 4, final product 1*3*7 = 21. 21 is not congruent to 4(mod 15), therefore 4 is composite.

Another way to state above theorem is, if divides , then n is prime.

 // CPP code to verify Vantieghem's Theorem  #include  using namespace std;     void checkVantieghemsTheorem(int limit)  {      long long unsigned prod = 1;      for (long long unsigned n = 2; n < limit; n++) {             // Check if above condition is satisfied          if (((prod - n) % ((1LL << n) - 1)) == 0)              cout << n << " is prime\n";             // product of previous powers of 2          prod *= ((1LL << n) - 1);      }  }     // Driver code  int main()  {      checkVantieghemsTheorem(10);      return 0;  }

Output:

2 is prime
3 is prime
5 is prime
7 is prime


The above code does not work for values of n higher than 11. It causes overflow in prod evaluation.

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