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Prime factors of 10...01

Prove that all prime factors \(p\) of \(1\underbrace{00\cdots 00}_{2015\text{ 0's}}1\) are of the form \(p\equiv 1\pmod{4}\).

Note by Daniel Liu
1 year, 6 months ago

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For contradiction, let \(p\equiv 3\pmod{4}\) satisfy \(\left(10^{1008}\right)^2\equiv -1\pmod{p}\).

Raise both sides by \((p-1)/2\) (which is odd); we get \(\left(10^{1008}\right)^{p-1}\equiv (-1)^{(p-1)/2}\equiv -1\pmod{p}\), which contradicts Fermat's Little Theorem. Mathh Mathh · 1 year, 6 months ago

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@Mathh Mathh This was the solution I was looking for. Good job! Daniel Liu · 1 year, 6 months ago

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Note \(n=100\cdots001=10^{2016}+1=(10^{1013})^2+1^2\). By Fermat's theorem on the sum of two squares, any prime factor \(p\) of \(n\) satisfies \(p\equiv1\pmod{4}\) or \(p=2\). Since \(n\) is odd, \(p\equiv 1\pmod{4}\). Maggie Miller · 1 year, 6 months ago

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@Maggie Miller Okay, I was hoping for a solution that didn't use Fermat's two square theorem. Thanks regardless. Daniel Liu · 1 year, 6 months ago

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@Daniel Liu I thought it was kind of clever! Haha, sorry that wasn't what you were looking for. Maggie Miller · 1 year, 6 months ago

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Note that number is (10^1008)^2+1. Since this is odd, the claim is true. Thinula De SIlva · 1 year, 6 months ago

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@Thinula De SIlva Do you mind explaining your solution a bit more? Daniel Liu · 1 year, 6 months ago

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@Daniel Liu Well Fermat said that if n is even, all prime factors of n^2+1 are 1mod4. Thus, we can see that 10^1008 is obviously even, so we're done. Thinula De SIlva · 1 year, 6 months ago

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