Prime Gaps

Number Theory Level 3

Is there a number $n$ such that there will never be a gap between two consecutive primes $p_{m}$ and $p_{m+1}$ greater than $n$ . That is to say $p_{m+1}-p_{m}<n$ for all values of $m$ ?

No, There is no such value of

n

Yes, a value of

n

can be shown to exist (though we don't know what it is) Yes, a value of

n

is known This is an open problem in mathematics (nobody knows)

1 solution

William Whitehouse
Apr 13, 2017

No value for n exists.

Imagine a set, S, of an arbitrarily large number of consecutive primes starting at 2: $\{2,3,5,7,...,p_{i}\}$ the product of this set will be an exceptionally large compound number, call this N.

N+1 may or may not be prime but N+2 is divisible by 2, N+3 is divisible by 3. This will work for any added value as compound numbers added on will be divisible by their constituent primes (i.e N+12 will be divisible by 2 and 3 but not necessary 4). This means that every number between N+2 and N+ $p_{i}$ is composite. Since $p_{i}$ can be made arbitrarily large there is no upper limit for n.

Prime Gaps

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