A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers, while numbers like 4 (2 × 2) are called composite. Central to number theory is the fundamental theorem of arithmetic, which states that every natural number greater than 1 is either prime or a unique product of primes. Euclid proved around 300 BC that there are infinitely many primes, and the prime number theorem, proven in the late 19th century, describes their statistical distribution, showing the probability of a large number being prime is inversely proportional to its number of digits. Several mysteries remain, such as Goldbach's conjecture (every even integer greater than 2 is the sum of two primes) and the twin prime conjecture (infinitely many prime pairs differing by 2). These numbers are vital in modern technology, particularly in public-key cryptography, which relies on the difficulty of factoring large numbers, and they also appear in abstract algebra as prime elements and ideals.