Prime numbers and their importance

In short: Prime numbers are natural numbers greater than 1 that are divisible only by themselves and 1. They form the basic building blocks of all natural numbers and are central to modern encryption.

In more detail: Every natural number can be uniquely represented as a product of prime numbers, known as prime factorisation.

In detail

• Infinity: The ancient Greeks already proved that there are infinitely many prime numbers.
• Prime factorisation: Every number greater than 1 can be uniquely broken down into its prime factors, the foundation of many mathematical proofs.
• Distribution: As numbers grow larger, primes become rarer, but never completely disappear.
• Primality tests: Efficient algorithms check whether a very large number is prime without trying all possible divisors.
• Encryption: RSA encryption relies on the fact that factoring large numbers into their prime factors is extremely difficult.
• Twin primes: Pairs of primes with a difference of 2, such as 11 and 13, raise mathematical questions that remain unsolved to this day.
• Largest known primes: Specialised projects use enormous computing power to search for ever larger prime numbers, usually Mersenne primes (kombinatorik-grundlagen, logarithmen-einfach-erklaert).