| Summary: | A hypothesis of the existence of perfect nonradiating modes in dielectric nanoparticles of an arbitrary shape is put forward. It is strictly mathematically proved that such modes exist in axisymmetric dielectric nanoparticles and have unlimited radiation <i>Q</i> factors. With the smart tuning of the excitation beams, perfect modes appear as deep minima in the scattered radiation spectra (up to complete disappearance), but at the same time, they have a substantial amplification of the fields inside the particle. Such modes have no analogs and can be useful for the realization of nanosensors, low threshold nanolasers, and other strong nonlinear effects in nanoparticles.
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