Prove that none of them is prime

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Homework Help Overview

The discussion revolves around proving that the numbers [n factorial + 2], [n factorial + 3], ..., [n factorial + n] are not prime for natural numbers n greater than or equal to 2. The original poster seeks assistance in understanding the problem and its implications regarding consecutive non-prime numbers.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the factors of the expressions involved, questioning how these factors relate to primality. There is discussion about the implications of factors like 2, 3, and n on the numbers in question.

Discussion Status

Some participants have offered hints regarding factors of the expressions, while others express confusion about the implications of these factors. There is ongoing exploration of how many consecutive non-prime numbers can be derived from the expressions, with various interpretations being discussed.

Contextual Notes

Participants are working under the constraints of the problem statement, which requires them to prove non-primality and deduce the existence of consecutive non-prime numbers without providing complete solutions. There is some uncertainty regarding the calculations and interpretations of the results.

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Homework Statement



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.


Homework Equations





The Attempt at a Solution



I really don't know how to do this question. Any help/hints would be appreciated.
 
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kmeado07 said:
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.

Hi kmeado07! :smile:

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


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.
 
Hi kmeado07! :smile:

(please use the !)
kmeado07 said:
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) :smile:
 


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?
 
kmeado07 said:
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.
kmeado07 said:
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!?
 


So there are n-2 consecutive non-primes starting with n! ?
 
kmeado07 said:
So there are n-2 consecutive non-primes starting with n! ?

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

yes, n-1 starting with n! + 2 …

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


you would start at n! + 1,000,003 ?
 
  • #10
kmeado07 said:
you would start at n! + 1,000,003 ?

Nowhere near …

try again :smile:​
 
  • #11


i don't know, I am confused!
 
  • #12


bump i'd like more info on this too
 
  • #13


kmeado07 said:
So there are n-2 consecutive non-primes starting with n! ?

tiny-tim said:
so if you wanted 1,000,000 consecutive non-primes, where would you start? :smile:

kmeado07 said:
you would start at n! + 1,000,003 ?
If n- 2= 1,000,000, what is n? It's that easy.
 
  • #14


tiny-tim said:
ooh, i got that slightly wrong, didn't i? :redface:

yes, n-1 starting with n! + 2 …

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

I really have no idea what's 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 don't see that, maybe I am not understanding the question..
 
  • #15


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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