“A prime number is a number divisible only by itself and one — and Euclid proved there are infinitely many of them around 300 BCE, a proof that still stands unchanged.”
Latin primus meant first — the superlative of prae (before). A prime number was Euclid's prōtos arithmos — first number, foundational number. In his Elements (around 300 BCE), Euclid defined a prime as a number measured by no number but a unit alone. He then proved, in Proposition 20 of Book IX, that prime numbers are more than any assigned multitude of prime numbers — there are infinitely many. The proof is among the most elegant in mathematics: assume there are finitely many primes, multiply them all and add 1, observe that the result is either prime or has a prime factor not in your original list. Either way, your list was incomplete.
The ancient Greeks were fascinated by primes because they seemed to have no pattern. The Sieve of Eratosthenes, developed around 240 BCE, was an algorithm for finding all primes up to a given number: write out the integers, cross off multiples of 2, then multiples of 3, then 5, and so on. What remains are the primes. The algorithm is still taught and still efficient for small ranges.
The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, describes the distribution of prime numbers using complex analysis. It has been neither proved nor disproved — it is the most famous unsolved problem in mathematics, and a $1 million prize awaits its solution. The primes' apparent randomness is still not fully understood more than 2,000 years after Euclid.
Modern cryptography depends entirely on primes. The RSA encryption system, developed in 1977, uses the difficulty of factoring the product of two large prime numbers as the basis for secure communication. Every encrypted website connection relies on prime numbers. Euclid's prōtos arithmos is the foundation of internet security.
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Every time you visit a secure website, you are using prime numbers. The padlock icon in your browser represents an RSA encryption system whose security depends on the impossibility of factoring a 2,048-bit number that is the product of two primes. Euclid did not know about the internet; he proved the theorem that makes it secure.
The Riemann Hypothesis remains open 165 years after Riemann proposed it. The primes' distribution along the number line is the most profound mystery in pure mathematics. They seem random but are entirely determined. The pattern exists; we cannot find it.
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