Fermat's Last Theorem for Amateurs, two questions

Question

Looking at Ribenboim's book, page 2.

First question: What is the difference in the notational use of $X, Y, Z$ versus $x, y, z$?

Second question: Why is this true: "If $n$ is odd then $X^n + Y^n = Z^n$ has a non-trivial solution if and only if $X^n + Y^n + Z^n = 0$ has a non-trivial solution."?

Answer 1

Upper-case letters are more variable than lower-case. The equation $X^n+Y^n=Z^n$ is to show the relationship among the variables. The lower-case letters $x,y,z$ indicate a specific solution.

$X$ is an indeterminate and $x$ is a number that gets plugged into it.

But really, you can ignore the difference.

Answer 2

First question. There is no difference. Ribenboim just chose upper case.

Second question. When $n$ is odd, $(-z)^n = -z^n$. So if $(x,y,z)$ satisfies $$ x^n + y^n = z^n $$ then $$ x^n + y^n + (-z)^n = 0 $$ so one of those equations has an integral solution just when the other does. You can study either.