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 the energy needs for any conceivable future of mankind on Earth. 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.

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.