How do we know light has no mass?

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Our understanding of the speed of light and many other things is predicated on the fact that light has no mass. As we can’t weight it directly like on a scale I am wondering (outside of mathematics) how we can test and prove this theory? Is it possible that light does have mass, it is just very very very small?

Further, if light has no mass, does it also have no energy? e=mc2 means energy for something massless would be 0. We know light has energy, so how does this equation work?

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