A theorem on quadratic residues

that is, the number of quadratic residues in the range 0 to (p-1)/2 exceeds the number of nonresidues in this range. This theorem seems to have been first conjectured by Jacobi and proved by Dirichlet [i]1 in connection with the theory of binary quadratic forms. Proofs are also given in the books of Bachmann [2 ] and Landau [3]. More recent proofs are due to Kai-Lai Chung [4] and A. L. Whiteman [5]. All known proofs, including the one given here, are analytic. While a really elementary proof would be of great interest, the following proof may merit consideration because of its brevity. Our starting point is the following Gaussian summation, proved in [3].