Infinitude of Primes

Euclid's theorem that no list of primes can be complete — there are infinitely many.

Specifications & Metrics

proved

c. 300 BC (Euclid, Elements IX.20)

Core Principles & Analysis

• Euclid's proof multiplies any finite list of primes, adds one, and finds a new prime among the factors.
• Euler's later proof sums the reciprocals of the primes, which diverge.
• The largest known primes are Mersenne primes, found by distributed volunteer computing.