>how can this be proved?:
>((5^125)-1)/((5^25)-1) is not a prime number.
>

Well, if it's a composite number it has factors. It's sufficient to find just one factor different from 1 and itself to establish that it's composite.

How do you do this for (5¹²⁵)-1)/(5²⁵-1)?

You may try some general formula. For example,

(5¹²⁵ - 1)/(5²⁵ - 1)

= 5¹⁰⁰ + 5⁷⁵ + 5⁵⁰ + 5²⁵ + 1


= (5⁵⁰ - Tau·5²⁵+1)(5⁵⁰ + Tau·5²⁵+1),

where Tau is the golden ratio. But this does not work, since Tau is irrational.

Or you may try checking some simple numbers like 3, 7, 9, 11, ... using modulo arithmetic.

For 7, powers of 5 modulo 7 are equal:

5⁰ = 1 (mod 7)
5¹ = 5 (mod 7)
5² = 4 (mod 7)
5³ = 6 (mod 7)
5⁴ = 2 (mod 7)
5⁵ = 3 (mod 7)
5⁶ = 1 (mod 7)
...

and then the sequence starts repeating itself with period 6 (= 7 - 1).

From here,

5⁰ = 1 (mod 7)
5²⁵ = 5¹ = 5 (mod 7)
5⁵⁰ = 5² = 4 (mod 7)
5⁷⁵ = 5³ = 6 (mod 7)
5¹⁰⁰ = 5⁴ = 2 (mod 7)

So the whole number is 18 or 4 (mod 7). Tough luck, it's not divisible by 7. But you may want to try other numbers.