Problems & Puzzles: Puzzles

 Puzzle 846. A follow up to Puzzle 844 Emmanuel asks for a Sudoku solution with the minimal quantity of distinct primes embedded in the 9 rows, 9 columns and two main diagonals, read in both directions. This is of course a follow up of the Puzzle 844 where he got the best good solutions looking for maximals. For this puzzle he sent the following start-up example solution.  I found a grid with only 63 primes:       3  2  7  4  9  5  1  8  6     4  9  5  1  8  6  3  2  7     1  8  6  3  2  7  4  9  5     2  7  3  9  5  4  8  6  1     9  5  4  8  6  1  2  7  3     8  6  1  2  7  3  9  5  4     7  3  2  5  4  9  6  1  8     5  4  9  6  1  8  7  3  2     6  1  8  7  3  2  5  4  9 But I do not know if this is minimal, writes Emmanuel. Q. Send your minimal solution.

Contributions came from Emmanuel Vantieghem, Dmitry Kamenetsky and Carlos Rivera

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Emmanuel wrote:

This is my minimal solution : 41 primes inside.

1 5 9 2 7 4 3 6 8
2 7 4 3 6 8 1 5 9
3 6 8 1 5 9 2 7 4
4 3 6 8 1 5 9 2 7
8 1 5 9 2 7 4 3 6
9 2 7 4 3 6 8 1 5
5 9 2 7 4 3 6 8 1
7 4 3 6 8 1 5 9 2
6 8 1 5 9 2 7 4 3

The primes are :
{2, 3, 5, 7, 13, 17, 23, 29, 31, 43, 47, 59, 67, 71, 89, 347, 367, 631, 743, 863, 887, 947, 1327, 1367, 1567, 2887, 4729, 5927, 18947, 23189, 26693, 28871, 34729, 65123, 156749, 947651, 3472951, 32756849, 48957631, 65123489, 96628871}

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Dmitry wrote:

44 primes:

819276543
762435198
354981627
543819276
198762435
627354981
276543819
435198762
981627354

primes 2 3 5 7 19 29 37 43 53 61 67 73 83 89 1567 1627 1987 4261 4561 4567 4987 7351 16249 16273 26153 26189 34267 56783 61837 67891 76243 76519 76543 89153 94261 94561 4267891 4537261 4987351 7354981 19876243 29156783 37894261 94261537

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Carlos Rivera wrote:

Does anybody may establish a theoretical lower limit for the quantity of primes inside a Sudoku according the rules of this puzzle?

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