<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] This is problem $87$ of Putnam and Beyond. Here is the solution:
$\dfrac{5^{125}-1}{5^{25}-1}=1+a+a^2+a^3+a^4$ where $a=5^{25}$.
We have $1+a+a^2+a^3+a^4+a^5=(a^2+3a+1)^2-(5^{13}(a+1))^2=(a^2+3a+1+5^{13}(a+1))(a^2+3a+1-5^{13}(a+1))$
The reason the second factor is larger than $1$ is $a^2=5^{50}>5^{39}>5^{13}(a+1)$

This problem was also part of the 1992 imo shortlist and was proposed by Korea, here is a link with the same solution but shorter: https://mks.mff.cuni.cz/kalva/short/soln/sh9216.html
</s>