Number Theory is the study of the integers. Integers are ...,-3,-2,-1,0,1,2,3,...
and are often noted as capital Z.
I'm reading three books on the topic this summer.
1) The Theory of Numbers by Niven, Zuckerman, and Montgomery
2) A Friendly Introduction to Number Theory by Silverman
3) The Higher Arithmetic by Davenport
The part I am reading about right now is all about divisibility.
So, what does it mean for a number to divide another number?
If we divide 100 by 25, then another way to think about is to imagine that we have 100 balls and we want to organize all 100 balls into piles of exactly 25 balls each. If we do this, how many piles of 25 will we have when we are done?
we can also think about dividing 100 by 25 by writing it as a fraction. 100/25 is 4. If we turn this fraction over and divide 25 by 100 then we get .25
25/100 = .25 which is not an integer but rather a decimal number. Anyway, what this means is we have 25 balls and we want to make 100 piles. If we make 100 piles then each pile can only have .25 of a ball in it.
Here is an example of a divisibility theorem from number theory:
if a divides b then a divides bc for any integer c
if a divides b then there is a number x such that ax = b. This is the definition of divisiblity.
lets take ax = b and multiply both sides by c
axc = bc
now, by the definition of divisiblity if ax = b where x is any integer, then a divides b.
So, let y = xc
we know that y is also an integer so ay = bc and so by definition, a divides bc
There are many more proofs like this one in number theory.