# Prove that none of them is prime (1 Viewer)

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1. The problem statement, all variables and given/known data

Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n].
Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the natural numbers.

2. Relevant equations

3. The attempt at a solution

I really don't know how to do this question. Any help/hints would be appreciated.

#### tiny-tim

Homework Helper
Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n].
Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the natural numbers.

Hint: what factor is there of (n! + 3)?

Re: Primes

Is (n factorial + 2) a factor of it?

Which would then apply to the others, which would show that none of them were prime.
For them each to be prime, their only factors would have to be themselves and 1.

#### tiny-tim

Homework Helper

Is (n factorial + 2) a factor of it?
No: n! + 2 is only one less than n! + 3 …

how can it be a factor of it?

Try writing n! + 3 in full (with n = 7, say)

Re: Primes

ok, so a factor of n!+3 is 3, and a factor of n!+2 is 2 and so on.
So a factor of n!+n is n. This shows that none of them is prime.

How would i go about doing the second part of the question?

#### tiny-tim

Homework Helper
Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the natural numbers.
ok, so a factor of n!+3 is 3, and a factor of n!+2 is 2 and so on.
So a factor of n!+n is n. This shows that none of them is prime.

How would i go about doing the second part of the question?
ok, so how many consecutive non-prime numbers are there starting with n!?

Re: Primes

So there are n-2 consecutive non-primes starting with n! ?

#### tiny-tim

Homework Helper
So there are n-2 consecutive non-primes starting with n! ?
ooh, i got that slightly wrong, didn't i?

yes, n-1 starting with n! + 2 …

so if you wanted 1,000,000 consecutive non-primes, where would you start?

Re: Primes

you would start at n! + 1,000,003 ?

#### tiny-tim

Homework Helper
you would start at n! + 1,000,003 ?
Nowhere near …

try again

Re: Primes

i don't know, im confused!

Re: Primes

#### HallsofIvy

Re: Primes

So there are n-2 consecutive non-primes starting with n! ?
so if you wanted 1,000,000 consecutive non-primes, where would you start?
you would start at n! + 1,000,003 ?
If n- 2= 1,000,000, what is n? It's that easy.

#### Firepanda

Re: Primes

ooh, i got that slightly wrong, didn't i?

yes, n-1 starting with n! + 2 …

so if you wanted 1,000,000 consecutive non-primes, where would you start?
I really have no idea whats going on starting with this

are you saying for n=10, then

2 + 10*9*8*...*2*1

has n-1 = 9 consectuve non primes?

because I dont see that, maybe im not understanding the question..

#### D H

Staff Emeritus
Re: Primes

Which numbers from 2 to 10 fails to divide 10! ? Given that, what divides 10! + 2? 10! + 3, and so on?

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