Necessary Center of Mass energy:

Ecm  =  mp  +  2 me

Laboratory energy and momentum:

Elab  =  Ep  +  pgamma

plab  =  pp  -  pgamma

Energy equation for the proton:

Ep = Sqrt[pp^2 + mp^2]

Equation to solve:

eq1  =  Elab^2 - plab^2  ==  Ecm^2

Solve:

pthreshold = pp /. Solve[eq1, pp][[1]]

Substitute energies (in eV); momentum of gamma is kT:

pthreshold /. {me ->  .511*10^6, mp -> 939. *10^6,  
        pgamma -> 2.7*8.617*10^(-5)}