## Primorials to Palindromes

The n-th primorial number, denoted   $p_n \, \#$   is defined as the product of the first   $n$   primes

$p_1 \, \# \; = \; 2$
$p_2 \, \# \; = \; 6$
$p_3 \, \# \; = \; 30$
$p_4 \, \# \; = \; 210$
$p_5 \, \# \; = \; 2310$

and the next few ones:

30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070

Let’s take the product of the first   $n$   primes and divide it by an integer $k$   in order to produce a palindrome

Starting with a few obvious solutions:

$11 \; = \; p_5 \, \# / (2\times 3\times 5\times 7)$   ……….   k = 210
$22 \; = \; p_5 \, \# / (3\times 5\times 7)$   ……………..   k = 105
$33 \; = \; p_5 \, \# / (2\times 5\times 7)$   ……………..   k = 70
$55 \; = \; p_5 \, \# / (2\times 3\times 7)$   ……………..   k = 42
$66 \; = \; p_5 \, \# / (5\times 7)$   ……………………   k = 35
$77 \; = \; p_5 \, \# / (2\times 3\times 5)$   ……………..   k = 30

Note that
$p_6 \, \# \; = \; 30030 \; = \; 2\times 3\times 5\times 7\times 11\times 13$,    and
$858 \; = \; 2\times 3\times 11\times 13$
The missing primes are 5 and 7, so   $k \; = \; 5\times 7 \; = \; 35$
So we write,    $858 \; = \; p_6 \, \# / 35$

Similarly,

$1001 \; = \; 7\times 11\times 13$
$1001 \; = \; p_6 \, \# / 30$,     $k \; = \; 30$

$2002 \; = \; 2\times 7\times 11\times 13$
$2002 \; = \; p_6 \, \# / 15$,     $k \; = \; 15$

$3003 \; = \; 3\times 7\times 11\times 13$
$3003 \; = \; p_6 \, \# / 10$,     $k \; = \; 10$

$5005 \; = \; 5\times 7\times 11\times 13$
$5005 \; = \; p_6 \, \# / 6$,     $k \; = \; 6$

$6006 \; = \; 2\times 3\times 7\times 11\times 13$
$6006 \; = \; p_6 \, \# / 5$,     $k \; = \; 5$

$595 \; = \; 5\times 7\times 17$
$595 \; = \; p_7 \, \# / 858$,     $k \; = \; 2\times 3\times 11\times 13 = 858$

$p_7 \, \# \; = \; 858\times 595$

$323 \; = \; 17\times 19$
[ Palindrome which is the product of two consecutive primes ]
$323 \; = \; p_8 \, \# / p_6 \, \#$
$p_6 \, \# \; = \; k \; = \; 2\times 3\times 5\times 7\times 11\times 13 \; = \; 30030$

$494 \; = \; 2\times 13\times 19$
$494 \; = \; p_8 \, \# / 19635$
$k \; = \; 3\times 5\times 7\times 11\times 17 \; = \; 19635$

$1346776431 \; = \; 3\times 7\times 11\times 13\times 17\times 23\times 31\times 37$
$1346776431 \; = \; p_{12} \, \# / 5510$
$k \; = \; 2\times 5\times 19\times 29 \; = \; 5510$

$35888853 \; = \; 3\times 7\times 11\times 13\times 17\times 19\times 37$
$35888853 \; = \; p_{12} \, \# / 206770$
$k \; = \; 2\times 5\times 23\times 29\times 31 \; = \; 206770$

Search for other palindromes with primorials   $p_n \, \#$    where n = 9, 10, 11

and   n > 12