Prove or disprove?

Mashimaru

Let $p_1 = 2, p_2 = 3, ...$ be the sequence of prime numbers. Prove or disprove the following statement: $\exists N,\forall n\geq N, p_n > 3n$

**Posted:** Sat Oct 17, 2009 2:32 am

PhilG · Poincare Conjecture Offline Joined: 24 Aug 2009 Posts: 173

All primes greater than 3 are either of form $6m - 1$ or $6m + 1$ otherwise they would be a multiple of $2$ or $3$ .
It can be checked that $p_9 = 29$ therefore $p_{9 + 2k} \ge 29 + 6k > 3(9 + 2k)$ and $p_{10 + 2k} \ge 31 + 6k > 3(10+2k)$ for $k \ge 0$
Therefore the assertion is true with $N = 9$

**Posted:** Sat Oct 17, 2009 5:19 am

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