Primorials to Palindromes

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

http://mathworld.wolfram.com/Primorial.html
https://en.wikipedia.org/wiki/Primorial

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
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged , . Bookmark the permalink.

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