BOMC-2012 Test Day 1 Problem 3

[*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 \[n\in N\]

[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.)

[Nadim Ul Abrar]

Yess ..
$mod 3$ , and $modp_1$ kills the problem ...

[*Mahi*]

I myself used $\bmod 6$ and $\bmod p_2$, as whatever $p_1$ is, $p_2$ must be odd.

[sm.joty]

I can't solve.
What's the process

[*Mahi*]

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 $p_x$.

[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 $3$