I think a proton needs 3 times the mass of an electron because a proton is 1836 times heavier than an electron but a neutron is 1839 times heavier than an electron.
proton + proton → neutron + proton + positron + neutrino
http://en.wikipedia.org/wiki/Proton-proton_chain_r...
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You are right that a neutron is more massive than a proton, so the proton needs to get extra mass from somewhere. The extra mass comes from the protons' kinetic energy. Remember Einstein's equation E=mc^2? (E is energy, m is mass, c is the speed of light). The kinetic energy of the protons is converted into extra mass to make the neutron, positron, and neutrino.
You may have heard of matter-antimatter reactions in which matter is destroyed, leaving behind only energy in the form of two photons. The proton to neutron reaction is the reverse. Energy gets converted into matter.