[Math] A question on consecutive prime numbers

number theoryprime numbers

Prime numbers:
$2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ ….

Difference between to consecutive primes:
$1$ $2$ $2$ $4$ $2$ $4$ $2$ $4$ $6$ ….

We know that there are infinite prime numbers. This is Ok. But does the difference between two consecutive prime numbers have any upper bound as the primes go to infinity? Can it be infinite?

Best Answer

Answer in the comments by David Mitra:

There are arbitrarily long sequences of successive integers, all of which are composite. (Start with n!+2.)

So no, the sequence does not converge.

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