Online Course Discussion Forum
MCIII Number Theory 2.10
Hello,
I need some help on 2.10. I don't know how to start, I tried listing out numbers like this:
3 2k-3
5 2k-5
7 2k-7
...
But I'm not sure how to proceed.
Thank you,
Tina Jin
You need to be careful reading the problem. We want to express an even number as the sum of two composite odd numbers. Composite odd numbers start with $9$, $15$, $21$, $25$, etc. A small even number often cannot be written as the sum of such two numbers. However, once the even number gets bigger, it is easier to do it, because there will be more composite odd numbers. To answer this quesiton, first find an even number that cannot be written as the sum of two composite odd numbers, and then prove every even number greater than that can be written that way.
I'm still a little bit stuck! I considered it mod 6, and considered cases of 2 mod 6, 4 mod 6, and 0 mod 6, and found a way to express the number as the sum of 25, 27, or 35 with an odd multiple of 3 for numbers greater than 38, and 38 cannot be expressed as the sum of two composite odd numbers and neither than any of the 19 smaller even numbers smaller than 38. I tested that 2, 4, ... 36 all don't work by hand, but how do I know without testing all those numbers. Is there a faster way to do this other than "bashing" all of the 19 smaller numbers?
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