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 (2
^{1}– 1) * (2^{2}– 1) = 1*3 = 3. 3 is congruent to 3 mod 7. We get 3 mod 7 from expression 3 * (mod (2^{3}– 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 <bits/stdc++.h> ` `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
"
``` `; ` ` ` ` ` `// 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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