Prove $\frac{5^{125}-1}{5^{25}-1}$ is not a prime

Asked Sep 30 '15 at 10:47

Active Sep 30 '15 at 11:35

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Prove $\displaystyle \frac{5^{125}-1}{5^{25}-1}$ is not a prime.

Some obvious thoughts:

$\displaystyle \frac{5^{125}-1}{5^{25}-1}={(5^{25})}^4+{(5^{25})}^3+{(5^{25})}^2+{5^{25}}+1$

UPD: A similar question: Prove that $\frac{ 5^{125}-1}{ 5^{25}-1}$ is a composite number

edited Apr 13 '17 at 12:20

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asked Sep 30 '15 at 10:47

For a prime $p$, $\phi(p)=p-1$ and it is equal to $(1-\frac1{p_1})(1-\frac1{p_2})...(1-\frac1{p_n})$ – Aditya Agarwal Sep 30 '15 at 10:52
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Maybe we need some factorization tricks to represent the formula instead of finding a specific prime which could divide it. – Sep 30 '15 at 10:55
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$a^n-b^n=(a-b)...$ – Aditya Agarwal Sep 30 '15 at 10:56
You need some trick because the factorization is $3597751×28707251×4032808198751×767186663625251×24687045214139234043375683501$ – lhf Sep 30 '15 at 11:00
@AdityaAgarwal Could we use $a^n-b^n=(a-b)\dots$ trick to ${5^{25}}^4+{5^{25}}^3+{5^{25}}^2+{5^{25}}+1$? – Sep 30 '15 at 11:00
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I think you should change those $5^{25}$ to $(5^{25})$. – barak manos Sep 30 '15 at 11:01
@barakmanos Edited. – Sep 30 '15 at 11:03
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http://math.stackexchange.com/questions/1345662/prove-that-dfrac-left-5125-1-right-left-525-1-right-is-composit – lab bhattacharjee Sep 30 '15 at 11:05
@labbhattacharjee Thanks! – Sep 30 '15 at 11:08
Strangely enough, it has $5$ prime factors. – barak manos Sep 30 '15 at 11:10

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