Difference between a hydrogen ion and a proton I've run into a bit of a problem on this weeks coursework.

A proton and an electron initially at rest combine to form hydrogen. Find the wavelength of the emitted photon? 

So, as far as I can see, there are two ways to do this: Find the mass of an electron + a proton, find the mass of hydrogen, work out Δm and use E=mc^2 and work out wavelength from E
Secondly, I'd assume, since I thought ionised hydrogen is the same as a proton, just use E=13.6 eV then work out wavelength from that. 
The problem is they give very different answers. 
Now all my friends are telling me to go with the first, so I reckon that's probably the right one. But why? Is a proton different to a H+ ion?
 A: The answers in the comments are both correct - a couple of points to add.
First think about the reverse process - how would you work out the energy required to break up the proton and electron in hydrogen to form a proton and electron. This is the same as the energy that needs to be released on formation of a hydrogen atom. I think you have figured this out from your question, but the point in the comment about it being easier to work things out this way is very good - the change in mass between H and a proton plus and electron is very small and this is not such a good way to work things out. 
Secondly, the question is not particularly well put. In reality the hydrogen atom could be formed with the electron in any level; n=1, 2, 3 etc. and the wavelength of the photon would be different in each case. The answers described in the comments and the methods you suggest both assume that you are forming a hydrogen atom in the ground state.
Finally $H^+$ is equivalent to a proton - unless you are thinking of a hydrogen nucleus of mass 2 with one proton and one neutron, but we normally write that as $D^+$.
