argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Euler's analytic proof

forThis argument, if it holds, bears in favour of the claim.constitution →There are infinitely many prime numbers.

The argument

If there were only finitely many primes, the sum of their reciprocals would be a finite sum and hence finite; because the sum of the reciprocals of the primes diverges, the primes cannot be finite in number.

Does it work?

Not yet evaluated.

Premises

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →The sum of the reciprocals of the prime numbers diverges.

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

Sep 16, 2026
The inference is valid: a finite set of primes would give a finite sum of reciprocals, so divergence rules out finiteness. The argument stands or falls with the divergence of the sum of prime reciprocals, Euler's theorem, which is itself settled with several independent proofs; the argument is an independent corroboration of Euclid's rather than a dependency of it.