Integers vary wildly in how "divisible" they are. One way to measure divisibility is to add all the divisors. This leads to 3 categories of whole numbers: abundant, deficient, and perfect numbers. We show there are an infinite number of abundant and deficient numbers, and then talk about what is known about perfect numbers. In particular, for even perfect numbers, each one corresponds to a Mersenne Prime.
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Number Theory is the study of whole numbers - also called integers. Since math began with the study of shapes and whole numbers, this is one of the oldest subjects around. However, it's also one of the most challenging. Many problems in number theory are easy to understand, but require a lot of cleverness to solve. It also does not require any advanced math to learn! Number Theory is for everyone.