prime's tree

Number Theory Level 2

$\begin{aligned}& 31\space\text{is prime.}\\& 331\space\text{is prime.}\\& 3331\space\text{is prime.}\\& 33331\space \text{is prime.}\\& 333331 \space\text{is prime.}\end{aligned}$ Does there exists such a large integer which is not prime in accordance to above observations?

Yes No

1 solution

Chris Lewis
Nov 12, 2019

Let $a_n=\underbrace{3 \cdots 3}_{n \; 3s}1$ . We have the recurrence relation $a_{n+1}=10a_n+21$ .

Simple divisibility tests show none of these numbers will be divisible by any of the primes $\{2,3,5,7,11\}$ . We can even show they're never divisible by $13$ . However, we can use the recurrence relation to easily investigate what happens to the remainders when $a_n$ is divided by $17$ :

$n$	remainder
$1$	$14$
$2$	$8$
$3$	$16$
$4$	$11$
$5$	$12$
$6$	$5$
$7$	$3$
$8$	$0$

so $a_8$ is divisible by $17$ .

prime's tree

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