Let A denote the set of all positive even integers. We call a number a in A “magic" if a cannot be expressed as a product of two other members of A.
I have found a general form for this series a=4x-y, where x is an int >1 and y is of the set {1,2,3}. I was wondering if there was a better way to write this?
Also how would we go about proving that every element of A can be expressed as a product of the "magic" numbers?
The magic numbers are all those that have only one factor of $2$ in their factorization, because when you multiply two even numbers you get two factors of $2$. These are the numbers equivalent to $2 \bmod 4$
Any member $a$ of $A$ that is equivalent to $0 \bmod 4$ can be factored as $2\cdot \frac a2$. Because it is a multiple of $4$, $\frac a2$ still has a factor $2$.
To prove any element of $A$ can be expressed as a product of magic numbers, factor it as $2^km$ with $m$ odd. You can then write it as the product of $k-1\ 2$'s and $2m$, all of which are magic.