Euler's formula is used to derive a number theory that groups natural numbers into four distinct sets obeying twenty new arithmetic rules of addition and multiplication.Mathematical definitions of composite numbers, prime numbers, odd numbers and even numbers are detailed.Collatz conjecture defines mathematical operations for even and odd numbers, which always converges to the unit value of one. It is solved by applying the even-odd operations, convergence formula, and unique operating nodes on the newly defined four natural number groups: leading to the final proof of the conjecture.
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