Each integer n>11 can be written as the sum of two composite numbers?

In summary, for any integer n greater than 11, it can be written as the sum of two composite numbers, regardless of whether n is even or odd. This is proven by considering two separate cases, where for case #1, n is even and for case #2, n is odd. By considering n-6 for case #1 and n-9 for case #2, it is shown that any integer greater than 11 can be written as the sum of two composite numbers.
  • #1
Math100
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Homework Statement
Establish the following statement:
Each integer n>11 can be written as the sum of two composite numbers.
[Hint: If n is even, say n=2k, then n-6=2(k-3); for n odd, consider the integer n-9.]
Relevant Equations
None.
Proof:

Suppose n is an integer such that ## n>11 ##.
Then n is either even or odd.
Now we consider these two cases separately.
Case #1: Let n be an even integer.
Then we have ## n=2k ## for some ## k\in\mathbb{Z} ##.
Consider the integer ## n-6 ##.
Note that ## n-6=2k-6 ##
=## 2(k-3) ##.
This means ## n=2(k-3)+6 ##
=## 2m+6 ##,
where ## m=k-3 ## is an integer.
Thus, ## 2m ## and ## 6 ## are two composite numbers.
Case #2: Let n be an odd integer.
Then we have ## n=2k+1 ## for some ## k\in\mathbb{z} ##.
Consider the integer ## n-9 ##.
Note that ## n-9=2k+1-9 ##
=## 2k-8 ##
=## 2(k-4) ##.
This means ## n=2(k-4)+9 ##
=## 2n+9 ##,
where ## n=k-4 ## is an integer.
Thus, ## 2n ## and ## 9 ## are two composite numbers.
Therefore, each integer ## n>11 ## can be written as the sum of two composite numbers.
 
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  • #2
Don't use ##n## for ##k-4##, you've already used it to mean ##2k+1##!
 
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  • #3
Yes, you're right, I shouldn't use n, I'll just use ## q ## then.
 
  • #4
For replacement.
 
  • #5
Other than that, is everything correct?
 
  • #6
Math100 said:
Other than that, is everything correct?
Yes. But "other than" is important. Never use the same variable name for two different variables.
 
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  • #7
Math100 said:
Therefore, each integer ## n>11 ## can be written as the sum of two composite numbers.
Where did you use the condition that ##n > 11##?
 
  • #8
Your idea works. For case #1 you may even consider ##n-4##. The ##n>11## requirement isn't really effective here. Suffices that ##n>6##.

For case #2, yes, it suffices to consider ##n-9## (because ##9## is the first odd composite on the list). One has ##2k+1-9 = 2(k-4)##. If ##n>11##, then surely ##k>5##.

Note also that this statement is optimised. You can't relax the premise to ##n\geqslant 11##, for example.
 
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FAQ: Each integer n>11 can be written as the sum of two composite numbers?

1. How do you prove that each integer n>11 can be written as the sum of two composite numbers?

The proof for this statement is known as Goldbach's Conjecture, which states that every even integer greater than 2 can be written as the sum of two prime numbers. Since every composite number can be expressed as the product of two or more prime numbers, this means that every even integer greater than 4 can be written as the sum of two composite numbers. Since n>11 is an even integer greater than 4, it follows that it can also be written as the sum of two composite numbers.

2. Are there any exceptions to this statement?

As of now, there are no known exceptions to this statement. However, since Goldbach's Conjecture has not been proven, it is possible that there may be some rare cases where an integer cannot be written as the sum of two composite numbers. Nevertheless, this has not been observed in any cases so far.

3. How does this statement relate to the Goldbach's Conjecture?

This statement is a variation of Goldbach's Conjecture, which specifically focuses on even integers greater than 4. By proving this statement, we are essentially providing further evidence for the validity of Goldbach's Conjecture.

4. Can this statement be extended to odd integers?

No, this statement only applies to even integers greater than 11. In fact, it has been proven that not all odd integers can be written as the sum of two composite numbers. For example, the odd integer 15 cannot be written as the sum of two composite numbers.

5. How does this statement impact number theory and mathematics as a whole?

This statement is a significant result in number theory, as it provides further insight into the distribution and properties of prime and composite numbers. The proof for this statement may also lead to further discoveries and advancements in the field of mathematics.

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