A180128 Maximal determinant of an n X n matrix whose elements are a permutation of the first n^2 prime numbers.

1, 2, 29, 6640, 4868296, 5725998504, 11305600374272, 35954639671827328

Offset: 0

Author

Hugo Pfoertner, Aug 11 2010

Keywords

• nonn
• hard
• more

Comments

The terms a(5), a(6), a(7) were found by tabu search, with strong numerical evidence for the optimality of a(7).

A known lower bound for the next term a(8) is 154665569137423060000.

Upper bounds for higher terms can be found by the method described by O. Gasper, H. Pfoertner and M. Sigg, and are given in A180127, e.g., a(8) <= 154715716383037989022.

An improved lower bound is a(8) >= 154671943501236284416, provided in a private communication by Richard Gosiorovsky. - Hugo Pfoertner, Aug 27 2021

Examples

			a(2) = 29:
. 7 3
. 2 5
a(3) = 6640:
. 23 11  5
.  3 17 13
.  7  2 19
a(4) = 4868296:
. 53 11 23 13
. 17 47 29  3
.  7  5 43 37
. 19 31  2 41
a(5) = 5725998504
. 89 41 23  2 53
. 31 97 29 47 11
. 59 13 79 61  7
. 37 19  5 83 67
.  3 43 71 17 73
a(6) = 11305600374272:
. 137  73   7  89  83  13
.  79 139  67  19   3  97
. 101   5 149  61  37  53
.   2 109 103  71 113  11
.  59  29  41  17 131 127
.  23  47  43 151  31 107
a(7) = 35954639671827332:
. 227  71 173  43  83  29  73
. 151 163   5 181   2 103  89
.  31 223 139  61 137  97  13
.  23  47 157 211 109  19 131
. 113   7  67 127 167 199  17
.  53  79 149  37  11 193 179
. 101 107   3  41 191  59 197
		

Links

• Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum JIPAM, Vol. 10, Iss. 3, Art. 63, 2008
• Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO]
• Index entries for sequences related to maximal determinants

Crossrefs

Cf. A180127 [upper bounds for a(n)], A085000 [maximal determinants for matrix elements 1, ..., n^2].

Cf. A340923, A340924, A340925.

Extensions

a(7) corrected, based on private communication from Richard Gosiorovsky by Hugo Pfoertner, Aug 27 2021

a(0)=1 prepended by Alois P. Heinz, Jan 19 2022