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 Post subject: Divisibility QuestionPosted: Sat, 12 May 2012 23:08:04 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
Find all positive integers , such that:

My Work:

I tried a lot of numbers.~

Clearly, is odd and not divisible by . Based on the numbers that I have tried I would like to conjecture that the only number is . Also, by Euler's Theorem, we note that the only prime that trivially satisfies this is . But, I don't see any way to show this for composite numbers.

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 Post subject: Re: Divisibility QuestionPosted: Sat, 12 May 2012 23:42:15 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 13970
Location: Austin, TX
rdj5933mile5math64 wrote:
Find all positive integers , such that:

My Work:

I tried a lot of numbers.~

Clearly, is odd and not divisible by . Based on the numbers that I have tried I would like to conjecture that the only number is . Also, by Euler's Theorem, we note that the only prime that trivially satisfies this is . But, I don't see any way to show this for composite numbers.

By the CRT you only need to check on prime powers, so you want to see for what prime powers, , but you know that , so you're saying that

In other words,

, i.e. .

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 Post subject: Re: Divisibility QuestionPosted: Sun, 13 May 2012 10:56:44 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6777
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
rdj5933mile5math64 wrote:
Find all positive integers , such that:

My Work:

I tried a lot of numbers.~

Clearly, is odd and not divisible by . Based on the numbers that I have tried I would like to conjecture that the only number is . Also, by Euler's Theorem, we note that the only prime that trivially satisfies this is . But, I don't see any way to show this for composite numbers.

By the CRT you only need to check on prime powers, so you want to see for what prime powers, , but you know that , so you're saying that

In other words,

, i.e. .

Hmm... does not give , but .

Anyway, the conjecture is obviously false with .

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 Post subject: Re: Divisibility QuestionPosted: Mon, 14 May 2012 15:08:31 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
outermeasure wrote:
rdj5933mile5math64 wrote:
Find all positive integers , such that:

My Work:

I tried a lot of numbers.~

Clearly, is odd and not divisible by . Based on the numbers that I have tried I would like to conjecture that the only number is . Also, by Euler's Theorem, we note that the only prime that trivially satisfies this is . But, I don't see any way to show this for composite numbers.

By the CRT you only need to check on prime powers, so you want to see for what prime powers, , but you know that , so you're saying that

In other words,

, i.e. .

Hmm... does not give , but .

Anyway, the conjecture is obviously false with .

Whoops . (I did but just for ... ) But doesn't work.

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math puns are the first sine of madness
-JDR

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