Infinitude of primes

What was discovered

Given any finite list of primes, the product of them all plus one has a new prime factor, so the primes are infinite. Together with the fundamental theorem of arithmetic, it is the starting point of number theory.

The same Book IX contains the characterisation of even perfect numbers.

What it led to

• analytic number theory
• the zeta function
• the prime number theorem
• public-key cryptography

Built on

• divisibility and the Euclidean algorithm

Made possible

• Sieve of Eratosthenesc. 240 BCE
• Euler product and zeta function1737
• Divergence of the sum of the reciprocals of the primesc. 1737
• RSA encryption1977

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