Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law probably used lattice point arguments. The details were supplied by several authors at the end of the 19th century (for references, see e.g. Hill's article below).

A modern approach using geometric ideas similar to those above was provided in several articles by R. Hill, such as this one.

Edit (2015). For reconstructions of Gauss's ideas see the recently published book Gauss's reciprocity laws in number theory (in German).

Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law probably used lattice point arguments. The details were supplied by several authors at the end of the 19th century (for references, see e.g. Hill's article below).

A modern approach using geometric ideas similar to those above was provided in several articles by R. Hill, such as this one.

Edit (2015). For reconstructions of Gauss's ideas see the recently published book Gauss's reciprocity laws in number theory (in German).

Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law probably used lattice point arguments. The details were supplied by several authors at the end of the 19th century (for references, see e.g. Hill's article below).

A modern approach using geometric ideas similar to those above was provided in several articles by R. Hill, such as this one.

Gauss's (unpublished and largely unknown) proof of the quartic reciprocity law probably used lattice point arguments. The details were supplied by several authors at the end of the 19th century (for references, see e.g. Hill's article below).

A modern approach using geometric ideas similar to those above was provided in several articles by R. Hill, such as this one.

Edit (2015). For reconstructions of Gauss's ideas see the recently published book Gauss's reciprocity laws in number theory (in German).