Integer and fractional self-imaging effects can be induced on periodic waveforms across the time, frequency , space or angular frequency domains, by imposing a quadratic phase profile along the corresponding Fourier dual domain. These quadratic phases should satisfy a strict condition, namely the well-known 'Tal-bot condition'. For fractional effects, the original period is divided by a natural factor, and the resulting self-image exhibits deterministic pulse-to-pulse phase variations that arise from the solution of a Gauss sum. This suggests the possibility of inducing self-imaging effects by imposing phase profiles that do not necessarily satisfy the Talbot condition. In this Letter, we show otherwise that the phase profiles retrieved from a Gauss sum also satisfy the Talbot condition. This finding indicates that the Talbot condition may encompass all possible quadratic phase patterns for inducing self-imaging effects. We establish here the precise relationships between the solutions of Gauss sums and the corresponding Talbot phases, and find out additional properties of Talbot phase patterns of fundamental and practical interest.