Nails are very mysterious. Whatever happens to them--infection, trauma, skin disease, or whatever--all they can mange to do is turn white or
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Yellow nail syndrome or crumbly. Often the only way doctors can tell what's going on is by context: Is there skin disease like psoriasis somewhere else on the skin? Are both feet involved? And so on.
The pattern you describe in yourself doesn't fit any pattern I'm familiar with, so I'm not surprised that your doctors didn't know what it was. Therefore, I have no way of knowing whether your son has the same thing you did, or something else. I think you should have him see a doctor who can check him for skin disease, fungus, or something else. It may turn out to be that he too has no idntifiable disease, but at least you'll know that there's nothing significant going on, other than the nails' appearance.
Best.
Dr. Rockoff