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WON plate
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[ April 17, 2002 ]
Substrings in Primes of Composite length.
by 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 21117 (11, 11, 17)
9719 (97, 71, 19)
1171 (11, 17, 71)
6 2, 3113131 (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 211111117 (11, 11, 11, 11, 11, 11, 17)
97973731 (97, 79, 97, 73, 37, 73, 31)
11317379 (11, 13, 31, 17, 73, 37, 79)
9 3113131373 (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, 51111971719 (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, 3113113113797
JCR [ April 20, 2002 ]
619719719719
JCR [ April 20, 2002 ]
nihil     JCR [ April 30, 2002 ]
14 2, 761311311131379
JCR [ May 15, 2002 ]
smallest = largest
JCR [ May 15, 2002 ]
nihil     JCR [ May 15, 2002 ]
15 3, 5nihil     JCR [ April 25, 2002 ]
nihil     JCR [ April 25, 2002 ]
nihil     JCR [ April 25, 2002 ]
16 21111111111111319
JCR [ April 19, 2002 ]
9797979797979719
JCR [ April 19, 2002 ]
nihil     JCR [ April 22, 2002 ]
18 2, 3113113113131131313
JCR [ May 15, 2002 ]
613131313131317971
JCR [ May 15, 2002 ]
nihil     JCR [ April 22, 2002 ]
20 2, 5nihil     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, 7nihil     JCR [ April 22, 2002 ]
nihil     JCR [ April 22, 2002 ]
nihil     JCR [ April 22, 2002 ]
22 2, 11nihil     JCR [ April 22, 2002 ]
nihil     JCR [ April 22, 2002 ]
24 2, 3113113113113113113137971
JCR [ May 19, 2002 ]
613131313131317971971971
JCR [ May 19, 2002 ]
25 51123911393133311393179399
JCR [ May 26, 2002 ]
9992333911939139313931397
JCR [ May 26, 2002 ]
?     
26 2, 13nihil     JCR [ May 27, 2002 ]
nihil     JCR [ May 27, 2002 ]


A000130 Prime Curios! Prime Puzzle
Wikipedia 130 Le nombre 130














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