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Old 2020-04-12, 14:04   #1
number0316
 
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Default One Theorem about Mersenne Numbers

All composite factors of prime-exponent Mersenne Numbers are strong pseudoprimes to base 2. However, I have a problem. How can I prove this theorem?
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Old 2020-04-12, 18:13   #2
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Quote:
Originally Posted by number0316 View Post
All composite factors of prime-exponent Mersenne Numbers are strong pseudoprimes to base 2. However, I have a problem. How can I prove this theorem?
MOD NOTE: Thread moved from "Abstract Algebra & Algebraic Number Theory"

This is elementary number theory.

Let p > 2 be a prime number. Every factor n > 1 of 2^p - 1 (whether prime or composite) is congruent to 1 (mod p). Therefore p divides n-1, so 2p - 1 divides 2n-1 - 1, so n divides 2n-1 - 1.

Let o be the odd part of n-1. Then p divides o. So 2o == 1 (mod n).

Last fiddled with by Dr Sardonicus on 2020-04-13 at 00:23 Reason: transcription error
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Old 2020-04-13, 04:23   #3
LaurV
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Grrr... couldn't you choose some other variable* instead of that "o"? Quite confusing with zero, (not only for beginners )

On the other hand, this is indeed more like for the homework subforum, it should be moved there.

----------
*On the funny side, I had a colleague in middle school who called all variables in equations "ț" (this is pronounced as the German "tz" group and at the time we were studying first and second degrees equations, and linear systems with few equations). He was good in math, and he wanted to follow a career in computer science, but when he found out that the keyboard has no ț (at the time, it was not possible to type ț, thingies like "word perfect" and graphic screens appeared years later), he went to physics. No joke. So, what started like a joke in the math class in 6th grade, because the teacher told us a variable can be anything, etc., and it was funny, and everybody was laughing when he was writing ț2+2ț+1=0 on the blackboard, ended up in changing his life forever.

Last fiddled with by LaurV on 2020-04-13 at 04:38
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Old 2020-04-13, 04:34   #4
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Quote:
Originally Posted by LaurV View Post
ț
https://en.wikipedia.org/wiki/T-comma

Even a manual typewriter could do that: T-Backspace-Comma

I don't see the problem there. It just needs a bit of engineering thinking.
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Old 2020-04-13, 04:44   #5
LaurV
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Quote:
Originally Posted by retina View Post
Even a manual typewriter could do that: T-Backspace-Comma
Oh your darkness, in spite of your avatar picture and your eternal wisdom, and you using non-JS very old stuff in your browser, you seem to be much too young to know a time when the computers didn't have a graphic screen, they even didn't use strange things called codepages, and there was no way to type on screen characters you don't have on the keyboard. And anyhow, the post says "keyboard", not "screen" or "paper". Or does your keyboard has a "ț" on it? (precious info which can be used to establish the location of the evil lair!)

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Old 2020-04-13, 04:57   #6
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Quote:
Originally Posted by LaurV View Post
Oh your darkness, in spite of your avatar picture and your eternal wisdom, and you using non-JS very old stuff in your browser, you seem to be much too young to know a time when the computers didn't have a graphic screen, they even didn't use strange things called codepages, and there was no way to type on screen characters you don't have on the keyboard. And anyhow, the post says "keyboard", not "screen" or "paper". Or does your keyboard has a "ț" on it?
I'm much too young to know a lot of things. Typewriters could micro-step the paper, they could print in two colours and the platter could move in two directions. With an appropriate amount of ingenuity almost any shape could be constructed. It sounds like these newfangled keyboards and screens you talk about are a backwards step from the mighty typewriters.

Last fiddled with by retina on 2020-04-13 at 04:58
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Old 2020-04-13, 05:42   #7
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You ably avoided my question... Probably you have seen the Raman text, I didn't hide it well enough

Edit: Re: Typewriter.

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Old 2020-04-13, 13:30   #8
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Quote:
Originally Posted by retina View Post
I'm much too young to know a lot of things. Typewriters could micro-step the paper, they could print in two colours and the platter could move in two directions. With an appropriate amount of ingenuity almost any shape could be constructed. It sounds like these newfangled keyboards and screens you talk about are a backwards step from the mighty typewriters.
Brain fart ?
The device on a typewriter that can move in 2 directions is a platen.

Last fiddled with by tServo on 2020-04-13 at 13:31
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Old 2020-04-13, 14:27   #9
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Quote:
Originally Posted by retina View Post
Even a manual typewriter could do that: T-Backspace-Comma
When writing a report for a chemistry class I was using a Panasonic KX-P1091 and was able to do that type of thing. I needed to write about sulphate and phosphate ions. I was using a simple text editor and a BASIC type-setting program. (It used slash commands inline in the text that were converted by a look up to escape codes.) I was able to produce stack subscripts and superscripts using the backspace command.
It looked more like SO{2-\atop 4} than SO42-
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Old 2020-04-13, 17:04   #10
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Quote:
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Brain fart ?
The device on a typewriter that can move in 2 directions is a platen.
Thank you, yes you are correct.
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Old 2020-04-14, 12:59   #11
number0316
 
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Quote:
Originally Posted by Dr Sardonicus View Post
MOD NOTE: Thread moved from "Abstract Algebra & Algebraic Number Theory"

This is elementary number theory.

Let p > 2 be a prime number. Every factor n > 1 of 2^p - 1 (whether prime or composite) is congruent to 1 (mod p). Therefore p divides n-1, so 2p - 1 divides 2n-1 - 1, so n divides 2n-1 - 1.

Let o be the odd part of n-1. Then p divides o. So 2o == 1 (mod n).
Thanks for the proof!
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