Ground-state filling of two spin-degenerate Landau levels ($E=\hslash (eB/m)[3N-eBA/(\pi \hslash )]$)

For $2N$ noninteracting electrons in a planar sample of area $A$, neglecting Zeeman splitting leaves two spin states for each orbital. Each Landau level therefore holds $D=eBA/(\pi \hslash )$ electrons. When $N\le D\le 2N$, fill the lowest level with $D$ electrons and the next with $2N-D$; summing their energies gives the displayed concave quadratic in $B$. The energies at the two integer-filling endpoints are equal.