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WON plate
130 |

[ 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 ?

 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 caseMore generally if L = 2 * k (i.e. even) and k >= 6, the third case(distinct substrings) can not exist. Indeed there are only 10 primeswith 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 ]

A000130 Prime Curios! Prime Puzzle
Wikipedia 130 Le nombre 130
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