Consecutive composite numbers — an example for RMO training

Find 20 consecutive composite numbers.

Consider the following


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


Consider the integers


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

(k+1)! + k


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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