One-Dimensional Crystals and Quadratic Residues Original Research Article

The main problem in crystallography is recovering the electronic density from the diffraction peak intensities. The one-dimensional model leads to recover a discrete Fourier series in imagenwith integral coefficients from its absolute value, which has arithmetical implications. In this paper we prove that the constant absolute value of Gaussian sums determines them among a class of exponential sums. This implies that if diffraction peak intensities are constant except for one of them, then, modulo translations, we obtain a quadratic residue molecule.