<s>[INST] Prove that $\frac{ 5^{125}-1}{ 5^{25}-1}$ is a composite number Prove that   $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that.
 [/INST] Put $x = 525$. Then $\frac{5^{125}-1}{5^{25}-1} = x^4+x^3+x^2+x+1 = (x^2+3^x+1)^2 - 5x(x+1)^2$ which has factor $(x^2+3x+1-513x-513)$ which is obviously $> 1$ and $< x^4+x^3+x^2+x+1$.
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