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[ November 6, 2002 ]
Double Cross Palindromes
by Enoch Haga (email)


Have a look at Enoch's following double cross squares
which are filled horizontally with different palprimes.
(131, 151 & 191)
(10301, 12421, 15451, 12721 & 18181)

131
151
191

10301
12821
15451
32323
18181

The 'double cross' refers to two additional requirements.
First, as indicated in the 3 x 3 example, the middle vertical bar
must be prime, preferably in both directions (359 & 953).
Secondly, as shown in the 5 x 5 example, the diagonals
must be palindromic and prime (ie. palprime)
and different from any horizontal one (12421).
Note that the center vertical of this 5 x 5 square
is only unidirectionally prime (38431)

Finally, the next 7 x 7 and 9 x 9 illustrations are the ones to beat !
No doubt there exist larger solutions to be discovered
and who knows with some extra hidden curiosities...

1242421
9110119
1579751
7507057
9174719
9127219
1489841

100111001
101424101
101414101
102202201
100888001
144202441
101292101
100050001
103696301

Of course the ultimate challenge would be to have all verticals
(not just the center) prime and in both directions. I expect that
someone will send us some of these.

[ November 8, 2002 ]
Already Jean Claude Rosa (email) contributes some
modest 3 x 3 solutions :


1 3 1           3 1 3
3 5 3           7 5 7
1 9 1           3 7 3
followed by this 5 x 5 square :

1 3 3 3 1
1 6 5 6 1
9 4 0 4 9
7 6 6 6 7
1 7 9 7 1

Note that all verticals are prime in both directions (a.k.a. emirps)!
(11971 & 17911)
(36467 & 76463)
(35069 & 96053)

[ November 15, 2002 ]
A triple 5 x 5 ALL ODD contribution from Jean Claude Rosa


1 9 9 9 1     7 1 9 1 7     9 3 7 3 9
3 3 5 3 3     9 7 5 7 9     1 7 9 7 1
7 7 3 7 7     7 9 3 9 7     1 5 5 5 1
1 3 9 3 1     7 7 9 7 7     9 7 3 7 9
1 9 3 9 1     7 1 3 1 7     9 3 1 3 9

[ November 22, 2002 ]
J. C. Rosa searched for a 5 x 5 grid embedded in a 7 x 7 grid
( there are no solutions if this 5 x 5 grid is 'all-odd' )
and he found only 3 solutions. Here is one :

3 3 3 1 3 3 3
3 7 7 3 7 7 3
9 9 3 2 3 9 9
7 3 9 2 9 3 7
3 9 3 1 3 9 3
1 7 9 3 9 7 1
3 3 3 7 3 3 3

Alas, all the verticals of the 7 x 7 grid are not emirps :

3397313 prime but 3137933 composite
3793973 palprime
3939373 prime but 3739393 composite
1322137 emirp

The interior 5 x 5 grid is complete
since all are "palprimes" or "emirps".

[ November 30, 2002 ]
This time J. C. Rosa constructed an 'all-odd' 7 x 7 grid,
complete, and embedded in an 'all-odd' 9 x 9 grid that is alas,
not complete !


9 9 7 7 3 7 7 9 9
1 3 7 9 9 9 7 3 1
1 1 5 7 9 7 5 1 1
3 3 7 9 3 9 7 3 3
1 1 3 1 1 1 3 1 1
9 1 7 9 3 9 7 1 9
7 7 5 1 9 1 5 7 7
1 3 9 1 3 1 9 3 1
9 7 3 3 3 3 3 7 9
911319719 composite but 917913119 prime
931311737 prime but 737113139 composite
775737593 prime but 395737577 composite
797919113 prime but 311919797 composite
399313933 emirp




A000142 Prime Curios! Prime Puzzle
Wikipedia 142 Le nombre 142














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