Heck, 350 operationally deployed would probably be more than enough.
You need 10,000 nukes when the enemy has hardened military infrastructure and when the enemy is multiplying and spreading with nukes all over the third world. The nuclear winter destruction of the world is a myth that Zerobamanites exploit to surrender to everything, including the drilling for oil and what not.
But all nuclear weapons are created equal. We have hundreds of tactical nukes with a yield of only a few dozen kilotons suitable for use on the battlefield to nukes in the megaton range for taking out submarines, hardened missile silos or command centers or entire cities.