Mathematical Proof that Photons have No Rest Mass

[Comeback City]

Using these equations I am about to prove that photons have a rest mass of zero (mathematically)
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E=hc/λ Photon Energy Equation

E²=(pc+mc²)² Mass-Energy Equivalence with Momentum Equation

p=h/λ Momentum of a Photon Equation

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Set the First 2 equations equal to each other...
E=hc/λ ------------------------------------------------> E²=(pc+mc²)²(hc/λ)²=(pc+mc²)²

h²c²/λ²=p²c²+m²c⁴

{h²c²/λ²}/c²={p²c²+m²c⁴}/c²

h²/λ²= (p²c²/c²)+(m²c⁴/c²)

h²/λ²=(p²)+(m²c²)

(h/λ=p+mc)²

[mc=(h/λ)-p]²

[mc(1/c)=(h/λ)(1/c)-p(1/c)]²

[m=(h/λc)-(p/c)]²

(Enter in the Momentum of a Photon Equation)

{m=(h/λc)-[(h/λ)/c]}²

[m=(h/λc)-(h/λc)]²

(m=0)²

m=0

m(photon) = 0

Is my math correct?
If so, is this legible?
And if so again, has this been proved yet and I am just completely unaware that it has?
I also made a video on YouTube about this if you want to check it out...

 

[mathman]

$(hc/λ)^2=(pc+mc^2)^2$

$h^2c^2/λ^2=p^2c^2+m^2c^4$

Second line does not follow from first, (right side is wrong).

 

[Comeback City]

Second line does not follow from first, (right side is wrong).

Thanks. I didn't see that until you pointed it out. So I just did the math again (the right way) and it comes out to be...

0 = m²c²

Could this also prove that the the mass of a photon is zero since we already have a set velocity for the speed of light (2.99x10⁸ m/s)?

 

[ZapperZ]

What you have done is simply proving a circular argument. For example

"p=h/λ Momentum of a Photon Equation"

is ALREADY an assumption that the momentum does NOT depend on ANY mass. In other words, you didn't show that this is true, but rather you adopt this form for the photon.

What you have done is simply doing a mathematical derivation from a known set of criteria, rather than showing that those starting criteria or description is correct. This is not a mathematical proof, but rather a logical consequence.

Zz.

 

[Comeback City]

What you have done is simply proving a circular argument. For example

"p=h/λ Momentum of a Photon Equation"

is ALREADY an assumption that the momentum does NOT depend on ANY mass. In other words, you didn't show that this is true, but rather you adopt this form for the photon.

What you have done is simply doing a mathematical derivation from a known set of criteria, rather than showing that those starting criteria or description is correct. This is not a mathematical proof, but rather a logical consequence.

Zz.

So in simple terms, are you saying that p=h/λ is an equation DERIVED from the other two equations (and possibly others) ?