In my imagination, nuclear power could disapear when we use up all the uranium generating electricity, or exploding nuclear bombs.
Is uranium a renewable resourse? if not, can we prove that this element could not be created by throwing neutrons and protons together?
235U is a nonrenewable resource in that when we use it in weapons or power plants it is not regenerated (and eventually after a long series of decay steps, mostly ends up as lead). However, there is a not entirely negligible amount being extruded by mid-oceanic rifts and volcanos, so there will continue to be a small supply that becomes physically reachable over extremely long timescales. (I wouldn't really count that as "renewable".)
You can create any element by colliding nuclei of lower-mass ones under appropriate circumstances, but doing so requires more energy than you would obtain from having a new fuel source, so to do that would be very unwise. Just use the electricity that you would have used to create 235U and use it directly.
Uranium is non-renewable in current methods of nuclear power generation, and there is only enough in the proven crust resources for a few hundred years. This is a serious problem, but it can be overcome by several methods.
Thorium is more plentiful than U235, and it can be used in a reactor. Modern ideas for thorium reactors are associated with Carlo Rubbia. Thorium breeds U233 from thorium, and U233 is fissile.
One can alternatively use thermonuclear weapons explosions to generate power. The fission component of these bombs are plutonium, which can be bred from regular Uranium using neutrons, which are produced during the fusion process. So thermonuclear power can make plutonium for triggers, and can add neutrons to thorium to make U233, which is also fissile. The fusion fuel itself is essentially infinite deuterium, and the neutron breeder can produce new fissile elements in sufficient amounts to make the whole thing self sustaning.
This approach to nuclear power is politically untenable nowadays, because of the risk of proliferation and theft associated to H-bombs. It is also an untried method, and will require deep underground cavities which are made hugely radioactive scattered throughout the world. Nevertheless, this is a complete solution to the problem of renewable energy. It is discussed in these questions: How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work? , Why was PACER abandoned? . It is an engineeringwise very practical way to achieve fusion power.
Is uranium a renewable resourse? if not, can we prove that this element could not be created by throwing neutrons and protons together?
No, Uranium is not a renewable resource, but this proposition lacks something more fundamental. Uranium in this context is an energy resource and not a material resource.
If you consider a natural material like diamond, through technology, we are capable of inventing new means of producing the material that can fully substitute the natural material. Obviously, if you can synthetically create something that you previously had to extract from the Earth, you can overcome the limitation of the finiteness of the material. However, this is only possible because the desired qualities of the diamond are things like hardness and luster. An energy resource is one where we only gain useful value from the release of the energy stored in it. Putting together the material from its constituents should produce a negative energy return on energy investment, and thus no assumptions about technology will give it value as an energy resource.
There is one proviso regarding Uranium in modern nuclear power. It's not 100% correct to label Uranium as an energy resource exclusively. This is because the value in Uranium lies predominantly in the Uranium-235 isotope which is the only naturally occurring stable fissile isotope. All fissions release nuclear energy, and there is a multitude of isotopes on Earth that are fissionable, and combined, these could support all energy needs for 10s of 1000s of years. In that sense, the nuclear fission energy resource on Earth is abundant and trivially accessible. The property of Uranium ore that we seek when we mine the substance is the neutron resource that comes from the U-235 isotope because the fissile property allows the nuclear energy resource to be exploited with relative ease. In current reactors it's not just the U-235 isotope that fissions (and thus releases energy), but the U-238 isotope is also transmuted by free neutrons and subsequently fissioned, accounting for nearly half of the energy over the plant's life-cycle. Other proposed reactor types and fuel cycles could allow an energy complex to operate sustainably with no new Uranium-235, simply by using the neutrons more efficiently.
