Number patterns in crystallography

Although the number patterns generated by the integer solutions of equations such as$\text{h}{}^{\mathrm{<mtext>2</mtext>}}\text{+k}{}^{\mathrm{<mtext>2</mtext>}}\text{+l}{}^{\mathrm{<mtext>2</mtext>}}\text{=n}$are of fundamental importance in crystallography, little work has been done to apply number theory, perhaps the most famous and fundamental area of mathematics, to problems involving the three-dimensional lattices on which crystal structures are based. The relevance of the work of such giants as Euler, Riemann, Dirichlet and Erdos to some crystallographic questions are discussed.