Do Photons Have Mass or Not?

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FE 26
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This may very well be an elementary question but I need to know once and for all. Do photons have mass and how is it proven? As of yet I have not received a sufficient answer to this question except for a vague explanation about how photons have mass until they cease movement, at which point they cease to be photons... I must be misunderstanding something somewhere because that doesn't quite make sense to me. Am I just being dense?
 
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In relativistic physics the main thing is not the mass but the total energy because it is what the particles exchange with. So photon has energy but has no mass in a usual sense, as the inertia coefficient in the Newton equation. Photons do not obey the Newton equation and they cannot stay at rest.