# Consecutive composite numbers — an example for RMO training

Find 20 consecutive composite numbers.

Consider the following

Theorem:

There are arbitrarily large gaps in the series of primes. Stated otherwise, given any positive integer k, there exist k consecutive composite integers.

Proof:

Consider the integers

$(k+1)!+2$

$(k+1)! +3$, and so on, so forth

$(k+1)! + k$

$(k+1)!+k+1$

Every one of these is composite because j divides $(k+1)!+j$, if $2 \leq j \leq k+1$.

Hence, for example, 20 consecutive composite numbers are as follows:

$20! +2$, $20! +3$, $20!+4$, $\ldots$, $20! + 21$.

More later,

Nalin Pithwa