Problems & Puzzles: Puzzles

Puzzle 731. Stanley Antimagic squares  of consecutive primes Natalia Makarova asks now for "Stanley Antimagic Squares of consecutive primes". Necessary definitions for the "Stanley Antimagic Squares" are given in the puzzle: http://www.primepuzzles.net/puzzles/puzz_681.htm She sent the solutions for n= 2 & 3: n=2, d=18 (minimal)   5 7  11 13   n=3, d=65573 (minimal)   21821  21839  21851 21841  21859  21871 21863  21881  21893 Q. Can you find solutions for n>3?

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Contributions came from J. K. Andersen

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Andersen wrote:

In puzzle 681 I found n=4 with minimal difference 82 between the 16 consecutive primes.
The first case is 4164532312868707261 added to each of the 16 numbers:
0  6 18 30
10 16 28 40
42 48 60 72
52 58 70 82

The second case is 6856521413120052187 added to these:
0 10 42 52
12 22 54 64
24 34 66 76
30 40 72 82

If the primes are minimized instead of their difference then there
are probably much smaller solutions but I haven't searched.

Later he added:

n=4, d=545468748354 (minimal)

136367186951 added to these:
 0  30  56  86
72 102 128 158
120 150 176 206
186 216 242 272

The first squares for n=4 start at 136367186951, 399926078933, 501929799281,
809511139667, 1038209011757, 1502332658587, 2351122716457, 2401736073493.

The smallest admissible width of a square for n=5 is 156 for
four patterns in http://dxdy.ru/post845503.html#p845503
None of them have an occurrence below 10^20 and finding 25 simultaneous
primes is infeasible. It also looks infeasible to find n=5 by generating
consecutive primes and testing them.

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