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 3 (Miller–Rabin)
- 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
- Primality Test | Set 4 (Solovay-Strassen)
- Primality Test | Set 2 (Fermat Method)
- Lucas Primality Test
- 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)
- 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)
- Midy's theorem
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
Improved By : SHUBHAMSINGH10