Prove that none of them is prime

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In summary: Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n].There are arbitrarily long finite stretches of consecutive non-prime nummbers in the natural numbers.
  • #1
kmeado07
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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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  • #2
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:
 
  • #3


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.
 
  • #4
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:
 
  • #5


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?
 
  • #6
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!?
 
  • #7


So there are n-2 consecutive non-primes starting with n! ?
 
  • #8
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:
 
  • #9


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?
 

1. What does it mean to prove that none of them is prime?

Proving that none of a set of numbers is prime means to show that none of them can be divided evenly by any other number except for 1 and itself, thereby eliminating the possibility of being prime numbers.

2. How do you prove that none of the numbers in a set is prime?

To prove that none of the numbers in a set is prime, you can use the divisibility test, also known as the Sieve of Eratosthenes, to eliminate all numbers that can be divided evenly by any other number. If there are no numbers left after this process, then none of them are prime.

3. Can a number be considered prime if it is only divisible by 1 or itself?

Yes, a number can be considered prime if it is only divisible by 1 or itself. This is the definition of a prime number - a number that is only divisible by 1 and itself.

4. Are there any exceptions to the rule that none of the numbers in a set is prime?

No, there are no exceptions to this rule. If a number can be divided evenly by any other number, it cannot be considered a prime number.

5. What is the significance of proving that none of the numbers in a set is prime?

Proving that none of the numbers in a set is prime can have various implications in different fields of study. In mathematics, it can help with identifying patterns and creating new theories. In computer science, it can aid in developing algorithms for prime number generation. In general, it helps in understanding the concept of prime numbers and their properties.

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