When $P_n=\frac{3^n-1}{2}$, prove that $P_{qn}$ is not a prime

Question

Let $P_n=\frac{3^n-1}{2}$, $q\in\mathbb N, q\neq 1$, then $$ P_q \text{ is composite } \Leftrightarrow \forall n\in\mathbb N: P_{qn} \text{ is composite}$$

In other words, each $q$ you find, such that $P_q$ is not prime (e.g. $q=2,5,11,17,19$) gives you a class $P_{qn}$, which is never prime.

Try to prove that if some prime number $p$ divides $P_q$ then it also divides $P_{qn}$. — Gerry Myerson, Jan 19 '14 at 20:12

score 1 · Answer 1 · answered Jan 19 '14 at 20:16

1

Hint. $ 3^n-1$ divides $3^{q n}-1.$

answered Jan 19 '14 at 20:16

Leox

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When $P_n=\frac{3^n-1}{2}$, prove that $P_{qn}$ is not a prime

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