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"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This reasoning does not hold. Momentum is the product of energy and velocity. Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $$c^2 p = E v$$ This holds for non-relativistic massive particles ($E = m c^2$ and therefore $p=mv$) as well as for massless particles like photons ($v = c$ and hence $p=E/c$).

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This reasoning does not hold. Momentum is the product of energy and velocity.

"How is this momentum defined (equations)?"

Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $$c^2 p = E v$$ This holds for non-relativistic massive particles (total energy dominated by rest-energy: $E = m c^2$, and therefore $p=mv$) as well as for massless particles like photons ($v = c$ and hence $p=E/c$).

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This is where your reasoning goes astray. Momentum is the product of energy and velocity. Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $c^2 p = E v$. This holds for non-relativistic massive particles ($E = m c^2$) as well as for massless particles like photons ($v = c$).

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This reasoning does not hold. Momentum is the product of energy and velocity. Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $$c^2 p = E v$$ This holds for non-relativistic massive particles ($E = m c^2$ and therefore $p=mv$) as well as for massless particles like photons ($v = c$ and hence $p=E/c$).

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This is where your reasoning goes astray. Momentum is the product of energy and velocity. This holds for photons as well as for massive particles. For massive non-relativistic objects energy is approximately equal to (invariant) mass, and therefore in that case (and only in that case) momentum is approximately equal to mass times velocity.

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This is where your reasoning goes astray. Momentum is the product of energy and velocity. Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $c^2 p = E v$. This holds for non-relativistic massive particles ($E = m c^2$) as well as for massless particles like photons ($v = c$).