0 = 0?

I used to hear some people said that photon is massless while has momentum. I was confused and surprising. I didn’t know why they said that.
Recently, I got their idea. It’s based on the equation below:
E² = p²C² + (m0C²)²
They assume m0 = 0 (photon is massless), then E = pC, or p = E/C (photon has momentum)
And according to Quantum, p = h/λ, then E = hC/λ = hγ
People can detect the energy of photon, so they can calculate the momentum of photon p.
It seems nice. But I think it carefully and found that it’s only a tricky mathematical game. My analysis is below:
Because the component p²C²and the component (m0C²)²are in equal position. If people can assume m0 = 0 while p ≠ 0, they can also assume p = 0 while m0 ≠ 0. That will be fair.
But a photon can not be both m0 = 0 and m0 ≠ 0 (or be both p ≠ 0 and p = 0 ), so the solution is m0 = 0, p = 0. It will be fair.
Then the equation E² = p²C² + (m0C²)² is actually talking 0 = 0. Or say, the frequency of the “photon” is zero (the wavelength is infinite). It does not exist.
Thank you.