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.
- 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.
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.
- Primality Test | Set 1 (Introduction and School Method)
- Primality Test | Set 2 (Fermat Method)
- Primality Test | Set 3 (Miller–Rabin)
- Primality Test | Set 4 (Solovay-Strassen)
- Lucas Primality Test
- AKS Primality Test
- Primality Test | Set 5(Using Lucas-Lehmer Series)
- Implementation of Wilson Primality test
- Primality test for the sum of digits at odd places of a number
- Chinese Remainder Theorem | Set 1 (Introduction)
- Wilson's Theorem
- Zeckendorf's Theorem (Non-Neighbouring Fibonacci Representation)
- Compute nCr % p | Set 2 (Lucas Theorem)
- Chinese Remainder Theorem | Set 2 (Inverse Modulo based Implementation)
- Combinatorial Game Theory | Set 4 (Sprague - Grundy Theorem)
- Compute nCr % p | Set 3 (Using Fermat Little Theorem)
- Using Chinese Remainder Theorem to Combine Modular equations
- Corollaries of Binomial Theorem
- Fermat's little theorem
- Nicomachus’s Theorem (Sum of k-th group of odd positive numbers)
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Improved By : SHUBHAMSINGH10