Summary: Prime Time
We can’t wait any longer! In this chapter we talk all about prime numbers.
- First, we define prime and composite numbers in Definition 6.1.1 and Definition 6.1.2. There is a lot of Prime fun to be had trying to find formulas for primes, or using Sage to compute.
- The foundational result enabling the rest of our usage of primes is Euclid’s proof of Infinitude of Primes, and the Sieve of Eratosthenes is a practical way to use this knowledge.
- We define prime factorization in Definition 6.3.1. Then the great theorem saying this is both always possible and unique is the Fundamental Theorem of Arithmetic. Some of the details of its proof are important on their own, such as Corollary 6.3.7.
- The following section gives many formulas that come directly as First consequences of the FTA.
- Finally, we make explicit the procedure for Converting to and from prime powers in solving congruences, along with several interesting examples such as Example 6.5.3 and Example 6.5.4.
In the Exercises, the ones that practice the conceptual basis of the Fundamental Theorem of Arithmetic are the best.
