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>Why does Hydrogen* only have one proton [and no neutrons]?
It doesn't; necessarily. Different isotopes of Hydrogen exist that have neutrons, such as Hydrogen-2 (Deuterium), Hydrogen-3 (Tritium; which is radioactive).
Elements are defined by their atomic number; the number of protons in an atom, if an atom has one proton it is hydrogen as hydrogen is defined as being the element with one proton. Also, Heroin (diamorphine) has 50 atoms.