Pep Talk

Number Theory is mainly about relations between rational numbers (fractions, which are denoted $\mathbf{Q}$).

Examples:

1. Every positive integer can be written in a unique way (up to order) as a product of primes.
2. The equation $x^2 = 2$ has no solution with $x\in\mathbf{Q}$.
3. The equation $y^2 = x^3 + 2$ has infinitely many solutions $(x,y)$ with $x,y\in\mathbf{Q}$. E.g.,
   
   $$(-1,1), \ldots, \left(\frac{66113}{80656}, \frac{36583777}{22906304}\right), \ldots$$
   
   
   
   
   $$\left(\frac{-64363752249455070879137307239023}{2937630569603... ...4957028437368992912415633092962882910696377832}\right), \ldots$$
   
   
   (In a few weeks you'll know how I found such big solutions.)

Much of number theory involves finding integer or rational solutions to polynomial equations. After all, solutions to such equations are relations between numbers.