Prove or disprove that \( \large \frac{5^{125}-1}{5^{25}-1}\) is prime.

*This problem is not original*

Prove or disprove that \( \large \frac{5^{125}-1}{5^{25}-1}\) is prime.

*This problem is not original*

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TopNewest\[\frac{x^5-1}{x-1} = \frac{(x-1)(x^4 +x^3 +x^2 +x +1)}{x-1} = 5^{100} +5^{75} +5^{50} +5^{25} +1 = y\] but \[3597751 | y \therefore\ not \ prime \] You also might be able to use the pseudo prime test i.e. \[a^{y-1} \neq 1 mod(y) \ for \ gcd(a,n) = 1 \] – Curtis Clement · 2 years, 3 months ago

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How did you guess that \(3597751 \mid y \)? Or did you use a computer? – Calvin Lin Staff · 2 years, 3 months ago

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– Curtis Clement · 2 years, 3 months ago

yea I used wolfram alphaLog in to reply

– Jihoon Kang · 2 years, 3 months ago

Haha. I guess this is a valid proof :) I proved it using difference of two squares.Log in to reply

– Curtis Clement · 2 years, 3 months ago

Can you share your solution pleaseLog in to reply

– Jihoon Kang · 2 years, 3 months ago

I already have :)Log in to reply

– Calvin Lin Staff · 2 years, 3 months ago

Great approach! Can you share your solution?Log in to reply

Let \(x=5^{25}\)

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

You can notice that \(x^4+x^3+x^2+x+1=(x^2+3x+1)^2-(5x^3+10x^2+5x)\)

\(=(x^2+3x+1)^2-5x(x^2+2x+1)\)

\(=(x^2+3x+1)^2-5^{26}(x+1)^2\)

And then, using difference of two squares, you get:

\(((x^2+3x+1)-5^{13}(x+1))((x^2+3x+1)+5^{13}(x+1))\)

So, the above expression has these two factors, hence it is not prime.

Q.E.D – Jihoon Kang · 2 years, 3 months ago

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– Shashank Rammoorthy · 2 years ago

Cool proof. How'd you get \(x^4+x^3+x^2+x+1=(x^2+3x+1)^2-(5x^3+10x^2+5x)\), though? Was it purely intuition?Log in to reply

– Jihoon Kang · 2 years ago

Yeah. I thought that difference of two squares may be the way to go, so I got \((Ax^2+Bx+1)^2\) and, through trial and error, found (fortunately) appropriate values for A and B so that it worked. It could have easily not worked either.Log in to reply

Nice. Here is a similar question :) – Calvin Lin Staff · 2 years, 3 months ago

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