Which number can I erase? All positive integers greater than $2$ are written on a board. First we erase number $3$ and $5$. With 4 positive integers $a,b,c,d$ satisfying $a+b=c+d$, if $ab$ is erased, then $cd$ can be erased, otherwise $cd$ cannot be erased. For example, $3=3 \times 1$, $3+1=4=2+2$ , then $2 \times 2 = 4$ is erased. a. What are the conditions of a number that can be erased ? b. If not only $3$ and $5$, but every prime number is erased at the beginning, can all other numbers be erased as well? If not, what are the conditions of a number to be erased ? (Sorry for my last question, English is my second language)

A prime $p>5$ can be erased if $m=2(p-1)$ has been erased, as $2+(p-1)=p+1$. Note that $$m=2\times(p-1)=4\times\frac{p-1}{2},$$ where $p-1$ is even because $p>5$ is prime. This factorization of $m$ shows that it can be erased if $\frac{p+5}{2}$ has been erased, because $$\frac{p-1}{2}+4=\frac{p+5}{2}+1,$$ where clearly $\frac{p+5}{2}5$.

A composite number $n=uv$ with $u,v>1$ can be erased if $m=u+v-1$ has been erased, as $$1+(u+v-1)=u+v.$$ Of course $m1$.

In particular, an integer $n>5$ can be erased if all integers less than $n$ have been erased, so you can use induction.