Primality Proof

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

This problem is not original

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

$\frac{x^5-1}{x-1} = \frac{(x-1)(x^4 +x^3 +x^2 +x +1)}{x-1} = 5^{100} +5^{75} +5^{50} +5^{25} +1 = y$ 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 - 6 years, 2 months ago

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 - 6 years, 2 months ago

yea I used wolfram alpha

Curtis Clement - 6 years, 2 months ago

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

Jihoon Kang - 6 years, 2 months ago

@Jihoon Kang – Great approach! Can you share your solution?

Calvin Lin Staff - 6 years, 2 months ago

@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 - 6 years, 2 months ago

@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 - 5 years, 11 months ago

@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 - 5 years, 10 months ago

@Jihoon Kang – Can you share your solution please

Curtis Clement - 6 years, 2 months ago

@Curtis Clement – I already have :)

Jihoon Kang - 6 years, 2 months ago

Nice. Here is a similar question :)

Calvin Lin Staff - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$