Problems & Puzzles: Puzzles

Puzzle 673 Sudoku & primes, again

My friend Jaime Ayala proposes a new puzzle related to Sudoku solutions & prime numbers (See puzzle 323):

• Let's suppose we have a "standard sudoku solution", SS
• Let's associate to each one of the nine integers in SS a distinct prime number from a set of nine consecutive primes in order to get the "transformed sudoku solution", TS
• Let's obtain the concatenation of these prime numbers in TS, by row (left to right) and by column (up to down), obtaining a total of 18 integers.
• How many primes are in these 18 integers?

I have computed the following valid example:

If the set of integers 1 to 9 is associated to the set of nine consecutive primes from 11 to 41, this is a valid couple of SS & TS:

 SS ST 5 1 2 9 8 6 3 7 4 23 11 13 41 37 29 17 31 19 3 4 7 2 1 5 6 8 9 17 19 31 13 11 23 29 37 41 6 8 9 3 4 7 2 5 1 29 37 41 17 19 31 13 23 11 8 5 6 7 2 9 1 4 3 37 23 29 31 13 41 11 19 17 1 2 3 4 5 8 7 9 6 11 13 17 19 23 37 31 41 29 7 9 4 1 6 3 5 2 8 31 41 19 11 29 17 23 13 37 9 7 1 8 3 2 4 6 5 41 31 11 37 17 13 19 29 23 2 3 5 6 9 4 8 1 7 13 17 23 29 41 19 37 11 31 4 6 8 5 7 1 9 3 2 19 29 37 23 31 11 41 17 13

BTW, in this valid couple of SS & TS we have zero primes in the 18 integers obtained by the asked concatenation!

Q. Get a SS with 18 primes in ST, or the best you can.

_____
You are able to use the set of 9 consecutive primes 11 to 41, or another set if you prefer.

Contributions came from Giovanni Resta, Emmanuel Vantieghem. Both found a  16 primes distinct example.

Giovanni claims that, based in his exhaustive approach (delivered on request), 16 is the maximum possible for consecutive sets of primes below 1000.

***

Giovanni wrote:

For Problem 673, I searched all the 9-primes consecutive ranges
below 1000 and the maximal number of primes in a Sudoku
scheme I found is 16. This was obtained for the 9-uples
starting at 11, 17, 19, 29, 31, 37, 41, 47, 59.

For example, for the numbers 11,13,17, 19,23,29, 31,37,41 I obtained:

6 1 5 7 2 3 4 9 8 =  29 11 23 31 13 17 19 41 37
2 9 4 1 6 8 7 5 3 =  13 41 19 11 29 37 31 23 17
7 3 8 5 9 4 2 6 1 =  31 17 37 23 41 19 13 29 11
4 2 6 8 3 1 9 7 5 =  19 13 29 37 17 11 41 31 23
8 7 3 4 5 9 1 2 6 =  37 31 17 19 23 41 11 13 29
9 5 1 2 7 6 8 3 4 =  41 23 11 13 31 29 37 17 19
5 4 9 6 1 2 3 8 7 =  23 19 41 29 11 13 17 37 31
1 6 2 3 8 7 5 4 9 =  11 29 13 17 37 31 23 19 41
3 8 7 9 4 5 6 1 2 =  17 37 31 41 19 23 29 11 13

All the 9 horizontal numbers are primes, plus
7 of the vertical ones, namely:

291331193741231117
114117133123192937
231937291711411331
311123371913291741
132941172331113719
173719114129133123
193113411137172329

***

Emmanuel wrote:

This is (up to now : my PC continues search to better ones) my best solution: 16 primes.

11  41  19  37  13  17  31  23  29
37  29  17  31  41  23  19  13  11
13  31  23  29  11  19  37  17  41
41  37  29  17  19  31  23  11  13
23  11  13  41  37  29  17  19  31
17  19  31  11  23  13  41  29  37
29  13  37  23  17  41  11  31  19
31  23  41  19  29  11  13  37  17
19  17  11  13  31  37  29  41  23
*    *

All rows give primes and all columns except those marked with * give primes !

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