Some interesting facts about Prime Numbers
- Except for 2, which is the smallest prime number and the only even prime number, all prime numbers are odd numbers.
- Every prime number can be represented in form of 6n + 1 or 6n – 1 except the prime numbers 2 and 3, where n is any natural number.
- 2 and 3 are only two consecutive natural numbers that are prime.
- Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
- Wilson Theorem: Wilson’s theorem states that a natural number p > 1 is a prime number if and only if
(p – 1) ! ≡ -1 mod p
OR,
(p – 1) ! ≡ (p-1) mod p
- Fermat’s Little Theorem: If n is a prime number, then for every a, 1 ≤ a < n,
an-1 ≡ 1 (mod n)
OR,
an-1 % n = 1
- Prime Number Theorem: The probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
- Lemoine’s Conjecture: Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture.