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

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