# induction?

• Sep 17th 2006, 09:41 PM
suedenation
induction?
Show that every integer greater than 11 is the sum of two composite integers.

Do I need to use induction for this question? I've tried, but it's not working.
Here's my work so far:

Let P(n): n=a+b for n>11, where a and b are composite integers.

Initial step: P(12) says that 12 is the sum of two composite integers because 12=4+8=6+6 where 4, 6 and 8 are all composite integers.

Induction step: Let k>11 and suppose P(k) is true and k=a+b.
We want to show that P(k+1) is true.
k+1=a+b+1

this is what i've tried, but actually I'm not sure if I'm on the right track. Just give me some suggestions. Thanks very much !!!:)
• Sep 18th 2006, 08:18 AM
ThePerfectHacker
Quote:

Originally Posted by suedenation
Show that every integer greater than 11 is the sum of two composite integers.

Do I need to use induction for this question? I've tried, but it's not working.
Here's my work so far:

Let P(n): n=a+b for n>11, where a and b are composite integers.

Initial step: P(12) says that 12 is the sum of two composite integers because 12=4+8=6+6 where 4, 6 and 8 are all composite integers.

Induction step: Let k>11 and suppose P(k) is true and k=a+b.
We want to show that P(k+1) is true.
k+1=a+b+1

this is what i've tried, but actually I'm not sure if I'm on the right track. Just give me some suggestions. Thanks very much !!!:)

Cool problem almost like the Goldbach conjecture.
---
Consider 2 cases for n>11.
~~~
n is even:
Then, for n=12 we can write,
n=4+8 both of which are composites and even.
Now, we prove that n+2 can be expressed as even composites.
Thus, for some n>11 even we have,
n=a+b where a and b are even composites then,
n+2=a+(b+2) where a is an even composite and (b+2)>b>2 is an even composite. Thus, this shows by induction a stronger case that any even number can be expressed as a sum of two even composites (not just composites) for n>11.
~~~
n is odd:
Then, for n>11 we can automaticall write,
n=9+(n-9) where 9 is a composite and,
n-9>2 because n>11 thus it is an even number greater then two and thus a composite.

Note: This was not completely a math induction problem only the part of it was.