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The Fast Folding Algorithm (FFA) is a phase-coherent search technique for periodic signals. It has rarely been used in radio pulsar searches, having been historically supplanted by the less computationally expensive Fast Fourier Transform (FFT) with incoherent harmonic summing (IHS). Here we derive from first principles that an FFA search closely approaches the theoretical optimum sensitivity to all periodic signals; it is analytically shown to be significantly more sensitive than the standard FFT+IHS method, regardless of pulse period and duty cycle. A portion of the pulsar phase space has thus been systematically under-explored for decades; pulsar surveys aiming to fully sample the pulsar population should include an FFA search as part of their data analysis. We have developed an FFA software package, riptide, fast enough to process radio observations on a large scale; riptide has already discovered sources undetectable using existing FFT+IHS implementations. Our sensitivity comparison between search techniques also shows that a more realistic radiometer equation is needed, which includes an additional term: the search efficiency. We derive the theoretical efficiencies of both the FFA and the FFT+IHS methods and discuss how excluding this term has consequences for pulsar population synthesis studies.