[ April 17, 2002 ]
Substrings in Primes of Composite length
Patrick De Geest
This WONplate will collect all the primes of composite length
such that any substring, with length determined by the prime factor(s) of this composite length, is prime as well.
Let us take the smallest prime of composite length 4
that fulfils the requirements i.e. 1117.
The prime factor(s) of 4 is (are) 2 so we only have to check
the substrings of length 2.
In so doing we see the three prime substrings 11, 11 and 17.
One further refinement, leading zeros are not allowed in the substrings. This happens already in the case of length 9 with prime 101137331. Prime factor(s) of 9 is (are) 3
so we have the following 3digit prime substrings 101, 011, 113, 137, 373, 733 and 331.
But 011 starts with a leading zero, so this solution is rejected.
The next valid smallest prime is 113131373 with
substrings 113, 131, 313, 131, 313, 137 and 373.
For this exercice I can distinguish three interesting
cases for each composite length.
The smallest prime
The largest prime
Smallest prime with all different prime substrings
I wonder... will there ever be a composite length
without a proper prime ?
JCR = Jean Claude Rosa (email)
Composite Length (A002808) Prime Factors  Smallest Prime  Largest Prime  Smallest Prime with Distinct Substrings 
4 2  1117 (11, 11, 17)  9719 (97, 71, 19)  1171 (11, 17, 71) 
6 2, 3  113131 (11, 13, 31, 13, 31) (113, 131, 313, 131)  617971 (61, 17, 79, 97, 71) (617, 179, 797, 971)  113173 (11, 13, 31, 17, 73) (113, 131, 317, 173) 
8 2  11111117 (11, 11, 11, 11, 11, 11, 17)  97973731 (97, 79, 97, 73, 37, 73, 31)  11317379 (11, 13, 31, 17, 73, 37, 79) 
9 3  113131373 (113, 131, 313, 131, 313, 137, 373)  997739773 (997, 977, 773, 739, 397, 977, 773)  113137337 (113, 131, 313, 137, 373, 733, 337) 
10 2, 5  1111971719 (11, 11, 11, 19, 97, 71, 17, 71, 19)(11119, 11197, 11971, 19717, 97171, 71719)  smallest = largest JCR [ April 20, 2002 ]  nihil JCR [ April 19, 2002 ] 
12 2, 3  113113113797JCR [ April 20, 2002 ]  619719719719JCR [ April 20, 2002 ]  nihil JCR [ April 30, 2002 ] 
14 2, 7  61311311131379JCR [ May 15, 2002 ]  smallest = largestJCR [ May 15, 2002 ]  nihil JCR [ May 15, 2002 ] 
15 3, 5  nihil JCR [ April 25, 2002 ]  nihil JCR [ April 25, 2002 ]  nihil JCR [ April 25, 2002 ] 
16 2  1111111111111319JCR [ April 19, 2002 ]  9797979797979719JCR [ April 19, 2002 ]  nihil JCR [ April 22, 2002 ] 
18 2, 3  113113113131131313JCR [ May 15, 2002 ]  613131313131317971JCR [ May 15, 2002 ]  nihil JCR [ April 22, 2002 ] 
20 2, 5  nihil JCR [ May 17, 2002 ]  nihil JCR [ May 17, 2002 ]  nihil JCR [ May 17, 2002 ] 
The third case More generally if L = 2 * k (i.e. even) and k >= 6, the third case (distinct substrings) can not exist. Indeed there are only 10 primes with 2 digits beginning 1, 3, 7, 9.JCR  [ May 22, 2002 ] 
21 3, 7  nihil JCR [ April 22, 2002 ]  nihil JCR [ April 22, 2002 ]  nihil JCR [ April 22, 2002 ] 
22 2, 11  nihil JCR [ April 22, 2002 ]  nihil JCR [ April 22, 2002 ] 
24 2, 3  113113113113113113137971JCR [ May 19, 2002 ]  613131313131317971971971JCR [ May 19, 2002 ] 
25 5  1123911393133311393179399JCR [ May 26, 2002 ]  9992333911939139313931397JCR [ May 26, 2002 ]  ? 
26 2, 13  nihil JCR [ May 27, 2002 ]  nihil JCR [ May 27, 2002 ] 
