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Primality Proof

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

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Note by Jihoon Kang
1 year, 9 months ago

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\[\frac{x^5-1}{x-1} = \frac{(x-1)(x^4 +x^3 +x^2 +x +1)}{x-1} = 5^{100} +5^{75} +5^{50} +5^{25} +1 = y\] 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 · 1 year, 9 months ago

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@Curtis Clement FYI For exponents with multiple digits, use { } for all of them to display properly, EG 5^{100}

How did you guess that \(3597751 \mid y \)? Or did you use a computer? Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin yea I used wolfram alpha Curtis Clement · 1 year, 9 months ago

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@Calvin Lin Haha. I guess this is a valid proof :) I proved it using difference of two squares. Jihoon Kang · 1 year, 9 months ago

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@Jihoon Kang Can you share your solution please Curtis Clement · 1 year, 9 months ago

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@Curtis Clement I already have :) Jihoon Kang · 1 year, 9 months ago

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@Jihoon Kang Great approach! Can you share your solution? Calvin Lin Staff · 1 year, 9 months ago

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@Calvin Lin Certainly

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 · 1 year, 9 months ago

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@Jihoon Kang 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? Shashank Rammoorthy · 1 year, 6 months ago

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@Shashank Rammoorthy 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. Jihoon Kang · 1 year, 6 months ago

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Nice. Here is a similar question :) Calvin Lin Staff · 1 year, 9 months ago

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