The most common isotope of hydrogen (protium) has no neutrons simply because it doesn't need them to be stable.
In heavier elements, multiple positively charged protons are packed tightly together in the nucleus. Because like charges repel, these protons would normally push each other apart. Neutrons act as a "glue" or buffer, separating the protons and adding the strong nuclear force to keep the nucleus from tearing itself apart.
Because an ordinary hydrogen atom only has a single proton, there is no proton-proton repulsion to counteract, making a neutron unnecessary for stability.
However, hydrogen can have neutrons. It has two other known isotopes:
Tritium: Contains one proton and two neutrons.
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