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BOMC-2012 Test Day 1 Problem 3

Posted: Wed Apr 11, 2012 11:07 pm
by *Mahi*
Determine if there exists an infinite sequence of prime numbers \[p_1, p_2,..., p_n,... \] such that \[|p_{n+1}- 2p_n|=1\]for each \[n\in N\]

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Thu Apr 12, 2012 11:52 am
by Phlembac Adib Hasan
There is no such a sequence.
Hint :
First start by $2$ and show it's impossible.Next think about the last digit of the primes.(Most of the problem was solved in this way.There were two remaining cases where I had to use Fermat's little theorem.)

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Thu Apr 12, 2012 1:09 pm
by Nadim Ul Abrar
Yess ..
$mod 3$ , and $modp_1$ kills the problem ...

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Thu Apr 12, 2012 7:13 pm
by *Mahi*
I myself used $\bmod 6$ and $\bmod p_2$, as whatever $p_1$ is, $p_2$ must be odd.

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Thu Apr 12, 2012 10:31 pm
by sm.joty
I can't solve. :(
What's the process :?:

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Fri Apr 13, 2012 12:20 am
by *Mahi*
sm.joty wrote:I can't solve. :(
What's the process :?:
If you take $\pmod 6$, you can show that if $p_2$ is of the form $6k+1$ or $6k-1$, then all $p_i$ must be of the form $6k+1$ or $6k-1$ respectively. Then try induction to get the general form of $p_x$.

Re: BOMC-2012 Test Day 1 Problem 3

Posted: Fri Apr 13, 2012 6:06 pm
by nafistiham
I used contradiction.
Firstly, made a system that must include such a sequence (if it existed!)
using $\bmod 3$ proved that every sequence in that system includes infinitely many multiples of $3$