What is the maximum even number that can not be expressed as sum of two composite odd numbers?

Consider the three odd composites $9,25,35$. These are, respectively, $0,1,2\pmod 3$. Thus, if $n$ is an even number, one of $n-9,n-25,n-35$ is an odd composite divisible by $3$ (well, supposing it is $>3$ at least). Thus $35+3=\fbox {38}$ is the largest even number that might be an example...inspection shows that it is indeed an example, hence the maximum example.


Hint: If $x\geq 18$ is divisible by $6$, then we can write $x$ as $(6n+3)+9$ for some positive integer $n$, where both $6n+3$ and $9$ are composite.

Can you do something similar if $x\equiv 2$ or $4\pmod 6$ and $x$ is large enough?

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Number Theory

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