Content
4: Number Theory

Number Theory

The Number Theory is a wonderful part of mathematics which studies how to solve various problems using properties that different numbers possess.

Motivation

As a motivation for studying, the following example might be good:

Is there a solution of x3+x2=y4+y+1x^3 + x^2 = y^4 + y + 1 for some integers xx, yy?

Aaaand ... No there is no solution due to the fact that both sides can't be at the same time:

odd=odd or even=even\text{odd} = \text{odd} \text{ or } \text{even} = \text{even}

Let us take two cases:

  1. y=eveny = \text{even}, then when we have y4+y+1y^4 + y + 1, then y4y^4 is as well even,. Now, adding (y4) even+(y) even=even(y^4) \ \text{even} + (y) \ \text{even} = \text{even}. Finally, ((y4+y)+1)((y^4 + y) + 1) results in even+odd\text{even} + \text{odd}, which is always odd\text{odd}.
  2. y=oddy = \text{odd}, then y4+y+1=odd+odd+odd=oddy^4 + y + 1 = \text{odd} + \text{odd} + \text{odd} = \text{odd}

So the LHS is always odd. Now, considering LHS, we can do a similar analysis:

  1. x=evenx = \text{even}, then x3+x2=even+even=evenx^3 + x^2 = \text{even} + \text{even} = \text{even}
  2. x=oddx = \text{odd}, then x3+x2=odd+odd=evenx^3 + x^2 = \text{odd} + \text{odd} = \text{even}

RHS is always even\text{even}. Here we arrive at a contradiction ⚡️ even≠odd\text{even} \neq \text{odd}, which proves, that there is no solution to the given equation.

This example reveals that by knowing how even and odd numbers react to adding them or raising to some power (in a sense knowing some theory behind these numbers), we can prove that there is no solution for a given equation. The example above is shown informally and serves rather as an intuition and motivation.

References

The following are some references to problem sets which were quite useful to get more practice with number theory. I think some of them were super useful during the exam preparation. In this chapter practice and experience really is the way to go:

CountabilityDivision and Divisors