Take any number and keep finding factors of that number that cannot be factored themselves. For example, 84 = 2 ⋅ 2 ⋅ 3 ⋅ 7, 455 = 5 ⋅ 7 ⋅ 13, or 897 = 3 ⋅ 13 ⋅ 23. These examples show that a number can be written as the product of prime numbers. This is called aprime factorization. A separate argument, that we will shortly get to, shows that this factorization is unique. This result has far reaching consequences and is called theFundamental Theorem of Arithmetic. This theorem shows that primes are the DNA of the number system. Essentially all of the results of number theory are theorems of the primes, the topic of this chapter.

1. 1.

   Just keep dividing until it is not possible to continue without having a remainder.

2. 2.

   Essentially it makes too many trivial exceptions in theorems in number theory.

3. 3.

   A consequence of the binomial theorem, see equation (2.21).

4. 4.

   A quick proof goes as follows: letx = ⌊x⌋ + r, where 0 ≤ r < 1. Then the inequality follows froma⌊⌊x⌋ + r⌋ = a⌊x⌋ and ⌊a(⌊x⌋ + r)⌋ < a⌊x⌋ + a.

Authors and Affiliations

1. (Home address), Beverly, MA, USA

   Randolph Nelson

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