Short answers, pulled from the story.
A prime number is a natural number greater than 1 that is not the product of two smaller natural numbers. For example, 5 is prime because the only ways to write it as a product, 1 times 5 or 5 times 1, involve 5 itself, while 4 is composite because it equals 2 times 2.
The number 1 is excluded from the primes and placed in its own category as a unit because counting it as prime would break key results. The fundamental theorem of arithmetic would fail, since every number could gain endless factorizations using copies of 1, and the sieve of Eratosthenes would eliminate all multiples of 1 and output only 1.
Euclid proved around 300 BC that the primes are infinite by showing every finite list of them is incomplete. Multiplying the listed primes together and adding one gives a number that leaves a remainder of one when divided by any prime on the list, so its prime factors must lie outside the list.
Goldbach's conjecture asserts that every even integer greater than 2 can be written as the sum of two primes. Christian Goldbach formulated it in a 1742 letter to Leonhard Euler, and it has been verified for large ranges of numbers but remains unproven.
Prime numbers are the basis of public-key cryptography, including the RSA cryptosystem and the Diffie-Hellman key exchange, with 2048-bit primes common. RSA relies on the difficulty of factoring large numbers into their prime factors, since multiplying two large primes is easy but recovering them from the product is hard.
The largest known prime is the Mersenne prime 2 to the power 136,279,841 minus 1, which has 41,024,320 decimal digits. It was found on the 12th of October 2024 by Luke Durant and the Great Internet Mersenne Prime Search, and since 1992 the largest known prime has always been a Mersenne prime.