Re: Why do protons not stick together without some neutrons?

This is a relatively advanced question  What you need for a *detailed*
answer of this question is to take the first month or two of a nuclear
physics course.  If you know the physics up to some basic quantum
mechanics, I reccomend getting a textbook like Krane's  Introductory
Nuclear Physics and reading at least the text of the chapters that
cover the properties of the nuclear force (4 and 5 in Krane do a decent job
of this, and you can skip the review of quantum mechanics at the start and
still understand these).  


What you need for an answer that fits here is to understand a few things
about the strong nuclear force.  It's strong and attractive at short range,
repulsive at very short range, and yes it has the property that it's
*nearly* independent of the charge of the nucleons involved.  But the
nuclear force is very complex, and quantum mechanical issues come into
play.  For the other three fundamental forces, physics has exact forms and
treatments.  There is no analytical form for the strong force, though there
are many models.  There's also a non-central tensor component of it, which
doesn't necessarily conserve orbital angular momentum.  Particles like
electrons don't feel the strong force at all.  


Physical systems (like the nucleus of an atom) will settle into their
lowest-energy state over time.  That time, for something as small as the
nucleus, may be incredibly fast on human timescales.  Because protons and
neutrons are different, adding them together at the same time can fill all
the distinct lowest energy levels of the nucleus.  If you start just piling
on protons, for example, the new ones immediately have to go to
higher-energy states (because of the Pauli principle), and you don't add
enough nucleons to create an attractive enough strong-force potential well
to keep everything bound.  So the system decays to separate protons, the
lowest-energy state.  If you add neutrons to the mix, they can fill their
own lower-energy levels and attract both each other and the protons, and
the nucleons (general term for protons and neutrons in a nucleus) aren't
forced into such high energy levels that it would be energetically
favorable for them to escape...so the system stays bound.  It's a simple
consideration (of a complex force) of the amount of energy binding the
system vs. what it has when it splits apart.  The same is true for a
neutron, left alone a neutron will simply decay into a proton, a neutrino,
and an electron.  But the extra binding energy of the nucleus can keep the
whole system together when they're bound up to other nucleons, because the
system's total energy might have to rise to allow the neutron to decay. 
That would be like a ball rolling up a hill spontaneously.  Energy would
have to come from somewhere for that to happen.


Even then, there are odd things to be seen in nature.  The diproton, for
example (2 protons), does exist...it just doesn't stay bound for long.  The
detailed nature of the answer to your question is very complex, and exact
models for nuclear binding and energy states are still on the forefront of
unanswered scientific questions.