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### Consecutive Non-Prime Integers

```
Date: 2/17/96 at 12:42:32
From: Anonymous
Subject: help in abstract algebra

For any integer n>1 there exist n consecutive non prime
integers. I have taken n=4 and then use 24,25,26,27 as a
specific example. However, for n large I don't see how this
can hold.  Can you please send me some insight on the general
case?
```

```
Date: 3/24/96 at 12:12:30
From: Doctor Steven
Subject: Re: help in abstract algebra

This is a tough problem but one that has a relatively simple
solution once you know it (until then it'll keep you stumped
for days).

Take a number n>1.  Then looking at a_k = (n+1)! + k + 1,
where k goes from 1 to n, you get:

a_1 = (n+1)! + 2,
a_2 = (n+1)! + 3,
:
:
a_n = (n+1)! + n + 1.

Notice we have n consecutive integers. Notice 2, 3, ...., n+1
all divide (n+1)!.  Notice also that 2, 3, (n+1) divide 2, 3, ...
(n+1) respectively.  So we have n consecutive composite
integers.

-Doctor Steven,  The Math Forum

```
Associated Topics:
High School Number Theory

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