A prime is any number of dots that refuses to form a rectangle
Line up six dots and they fit neatly into two rows of three. Try the same with five or seven and nothing works except a single line. That is the essence of a prime: a whole number above 1 that no two smaller whole numbers multiply to make. Everything else above 1 is composite.
Put another way, a prime has exactly two divisors, 1 and itself, which is why 1 is excluded. Among 1 to 6, only 2, 3 and 5 qualify, since 4 is 2 times 2 and 6 is 2 times 3. Every prime except 2 is odd, and in ordinary decimal writing every prime above 5 ends in 1, 3, 7 or 9, because other final digits signal divisibility by 2 or 5. There are 25 primes below 100. Their importance comes from the fact that every larger whole number breaks down into primes in essentially one way.
Testing primality is a practical problem. Trial division, checking for divisors up to the square root, is simple and slow. The Miller–Rabin test is quick but carries a small risk of error, while the AKS test is always right in polynomial time yet too sluggish to be practical. Special forms such as Mersenne primes allow very fast tests and have produced record-breaking large primes. Public-key cryptography depends on how hard it is to split big numbers into prime factors.
The Rhind Mathematical Papyrus of around 1550 BC treats fractions with prime and composite denominators differently, but the earliest real study of primes is Greek. Euclid's Elements, around 300 BC, proves there are infinitely many and links Mersenne primes to perfect numbers, and the Sieve of Eratosthenes still generates prime lists. Around 1000 AD Ibn al-Haytham found Wilson's theorem, Ibn al-Banna' al-Marrakushi sped up the sieve, and Fibonacci's Liber Abaci of 1202 first described trial division.
Fermat stated his little theorem in 1640, and Christian Goldbach proposed in a 1742 letter to Euler that every even number is a sum of two primes. Euler brought analysis to bear, showing that the reciprocals of the primes add up without limit. Pafnuty Chebyshev proved Bertrand's postulate in 1852, and Riemann's 1859 ideas led to the prime number theorem in 1896. Goldbach's conjecture and the twin prime conjecture, that primes two apart never run out, remain open.
Source: Prime number
