math theory- primes problem

Question

I have tried to prove it for over $15$ hours with no success. I got a clue to use the following technique: between $((P_n), \:2(P_n))$ there is an additional prime hiding there - $P_{n+1}$.

please only clues and no solutions!

Thanks.

score 4 · Accepted Answer · answered Dec 19 '15 at 11:07

4

Given the hint you got, you know that $p_{n+3} \le 2 p_{n+2} \le 2 (2p_{n+1})$. Thus $p_{n+3}^2 \le 8 p_{n+1}p_{n+2} < p_n p_{n+1}p_{n+2}$ for all $n$ such that $p_n > 8$. That is $n \ge 5$.

You can check by hand it also holds for $n=3$ and $n=4$.

answered Dec 19 '15 at 11:07

quid

42,135

quid, perfect :) – Paolo Leonetti Dec 19 '15 at 11:07
Thanks. Not sure it is perfect though. Maybe one should avoid the special casing. – quid Dec 19 '15 at 11:10
2

I asked for a clue.. anyway thanks this solution is very nice. – Stav Alfi Dec 19 '15 at 11:16
@StavAlfi yes, sorry, I did not really register this. Else I might have left out a detail here or there. – quid Dec 19 '15 at 11:17

math theory- primes problem

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