Math · 3
What did Euclid's classic argument actually establish about prime numbers, as opposed to a direct claim that infinitely many exist?
Answer
That beyond any finite list of primes, at least one more prime must exist
Euclid's proof (Elements, Book IX, Proposition 20) never states 'there are infinitely many primes' outright; it shows that for any finite list of primes, a new prime not on that list can always be found, which logically implies an endless supply without ever invoking an actual infinite set.
💡 Did you know?
Euclid never actually proved infinitely many primes exist—he only showed that for any finite list, one more exists beyond it. A crucial logical distinction.
Other options people guess
- That there are exactly infinitely many primes, stated as a direct fact
- That the sum of the reciprocals of all primes diverges to infinity
- That every prime number can be written as the sum of two smaller primes
Topics
- primes
- euclid
- proof-theory
- number-theory
- history-of-math
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