All positive integers can appear on the board. It is sufficient to show that the following two claims $ ( 1) ,( 2)$ are true : $ ( 1)$ For $ k\in \mathbb{N}$, $ k$ can be written if and only if $ k^{2}$ can be written. $ ( 2)$ For $ k\in \mathbb{N}$, $ k^{2}$ is written on the board, then $ ( k+1)^{2}$ can be written. Since $ k^{4} +k^{2} +1=\left( k^{2} +k+1\right)\left( k^{2} -k+1\right)$, $ ( 1)$ is clearly true. Suppose that $ k^{2}$ is written on the board. Note that $ k^{4} +( k+1)^{4} +1=\left( k^{2} +( k+1)^{2} +1\right)\left( k^{2} +k+1\right)$. Therefore, $ ( k+1)^{2}$ can be written on the board, and so $ ( 2)$ is true. $ \blacksquare $