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 Post subject: no primesPosted: Wed, 18 Jan 2012 15:28:48 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7623
Location: NCTS/TPE, Taiwan
A easy one:

(1) Prove that there are no primes in the sequence 10001,100010001,1000100010001, ...
(2) Prove that, for every natural number n>1, there are only finitely many primes in the sequence
(3) What is the minimum number of terms from the sequence you need to test to guarantee the sequence contains no primes?

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 Post subject: Re: no primesPosted: Thu, 19 Jan 2012 10:04:39 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15562
Location: Austin, TX
There don't seem to be many takers, I hope you don't mind outermeasure, but for those mulling it over,

Hint:

Spoiler:
This really amounts to some long division, and counting becomes easy once you figure out why there are only finitely many (at least if you do it like I did).

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 Post subject: Re: no primesPosted: Thu, 19 Jan 2012 11:06:40 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7623
Location: NCTS/TPE, Taiwan
There don't seem to be many takers, I hope you don't mind outermeasure, but for those mulling it over,

Hint:

Spoiler:
This really amounts to some long division, and counting becomes easy once you figure out why there are only finitely many (at least if you do it like I did).

Indeed that's my approach too.

Spoiler:
It is secretly an algebra question rather than a number theory question.

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 Post subject: Re: no primesPosted: Thu, 31 May 2012 20:48:42 UTC
 S.O.S. Oldtimer

Joined: Fri, 27 Jul 2007 10:17:26 UTC
Posts: 279
Location: Chandler, AZ, USA
I was working on another similar problem: trying to prove is never prime for , when I remembered this problem that I didn't have time to look at a few months ago.

To answer this, consider where .

Note that , so is composite whenever is.

Also as a geometric series with .

So .

If is odd, this equals

and so is composite when is odd (in particular, when is an odd prime).

So the contains either one or zero primes, depending on whether is prime.

For is composite, so the sequence contains no primes.

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