Number Theory Observation

I was doing some number theory problems and while working on a problem I came up with an useful generalization. I know it is not very special but it's worth sharing in my view.

Note by Arghyadeep Chatterjee
9 months, 1 week ago

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Nice result! However, I think you can greatly improve your writing and formatting; it seems very hard to read and follow, but I would probably chalk that down to your possible lack of experience with mathematical writing.

A Brilliant Member - 9 months, 1 week ago

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I know but I cannot figure out how to use Latex .(I have absolutely 0 coding experience as I had never taken up computer science.) I try to use it but the writing gets all messed up. Maybe someone will type it out on Latex.

Arghyadeep Chatterjee - 9 months, 1 week ago

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I'm not just talking about LaTeX (I can understand that you may not know how to use it); I found the wording of the proof quite hard to follow. For example, instead of saying lines 2 to 7, you could say the following:

"As 22 and aa are coprime, we would like to find xanx \equiv a^n (here, the "\equiv" sign means "defined to be") such that

x=1(mod 2) x = 1 \quad (\text{mod } 2)

and

x=0(mod a). x = 0 \quad (\text{mod } a).

We note that this solution is unique by the Chinese Remainder Theorem."

If you were using LaTeX, you'd also want to take advantage of the Theorem/Proof layout, but even if you don't you would need to separate the theorem from the result. The subsequent six lines could also be rewritten in a similar flavour.

A Brilliant Member - 9 months, 1 week ago

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@A Brilliant Member Okay understood. I will try to better it ................You know after we've kicked our asses in cricket :)

Arghyadeep Chatterjee - 9 months, 1 week ago

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@Arghyadeep Chatterjee I hate the Aussie cricket team. National disgraces.

A Brilliant Member - 9 months, 1 week ago

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@A Brilliant Member I prefer rugby myself. I am a huge fan of Dave Pocock and Ben Foley

Arghyadeep Chatterjee - 9 months, 1 week ago

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Somewhat more directly, it just boils down to a2a(mod2a) a^2 \equiv a \pmod{2a} . This follows since a(a1)0(mod2a) a (a-1) \equiv 0 \pmod{2a} as the first term is a multiple of aa and the second term is a multiple of 2.

Calvin Lin Staff - 9 months ago

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