4 - Under Siege

Summary

Monday – January 2, 1792

I begin the new year with more determination and a renewed resolve to study prime numbers. One of my goals is to acquire the necessary mathematical background to prove theorems.

Prime numbers are exquisite. They are whole pure numbers, and I can manipulate them in myriad ways, as pieces on the chessboard. Not all moves are correct but the right ones make you win. Take, for example, the process to uncover primes from whole numbers. Starting with the realization that any whole number n belongs to one of four different categories:

The number is an exact multiple of 4 : n = 4k

The number is one more than a multiple of 4 : n = 4k + 1

The number is two more than a multiple of 4 : n = 4k + 2

The number is three more than a multiple of 4 : n = 4k + 3

It is easy to verify that the first and third categories yield only even numbers greater than 4. For example for any number such as k = 3, 5, 6, and 7, I write: n = 4(3) = 12, and n = 4(6) = 24; or n = 4(5) + 2 = 22, and n = 4(7) + 2 = 30. The resulting numbers clearly are not primes. Thus, I can categorically say that prime numbers cannot be written as n = 4k, or n = 4k + 2. That leaves the other two categories.