[ 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 3-digit 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	113113113797 JCR [ April 20, 2002 ]
	619719719719 JCR [ April 20, 2002 ]
	nihil     JCR [ April 30, 2002 ]
14 2, 7	61311311131379 JCR [ May 15, 2002 ]
	smallest = largest JCR [ 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	1111111111111319 JCR [ April 19, 2002 ]
	9797979797979719 JCR [ April 19, 2002 ]
	nihil     JCR [ April 22, 2002 ]
18 2, 3	113113113131131313 JCR [ May 15, 2002 ]
	613131313131317971 JCR [ 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	113113113113113113137971 JCR [ May 19, 2002 ]
	613131313131317971971971 JCR [ May 19, 2002 ]
25 5	1123911393133311393179399 JCR [ May 26, 2002 ]
	9992333911939139313931397 JCR [ May 26, 2002 ]
	?
26 2, 13	nihil     JCR [ May 27, 2002 ]
	nihil     JCR [ May 27, 2002 ]