[seqfan] Squares with prime sums in rows and columns

zak seidov zakseidov at yahoo.com
Thu Oct 22 16:55:57 CEST 2009

```Dear seqfans,

is it new/interesting :-))
or old/dumb :-(

Thanks, Zak

Lexicographically least NxN squares of N^2 subsequent integers (not necessarily starting with n=1) with prime sums in each row and each column
==============================================

1x1 square
rowsums={2}, colsums={2}
2
=

2x2 square
rowsums={3,7}, colsums={5,5}
1,2
4,3
==

3x3 square
rowsums={7,19,19}, colsums={13,13,19}
1,2,4
5,8,6
7,3,9
=====

4x4 square
rowsum={11,29,43,53}, colsum={23,31,41,41}
01,02,03,05
04,06,07,12
08,09,15,11
10,14,16,13
===========

5x5 square
rowsums={17,41,67,97,103}, colsums={53,53,59,71,89}
01,02,03,04,07
05,06,08,09,13
10,11,12,14,20
15,16,19,23,24
22,18,17,21,25
==============

6x6 square
rowsums={23,59,89,131,167,197}, colsums={97,103,109,113,113,131}
01,02,03,04,05,08
06,07,09,10,11,16
12,13,14,15,17,18
19,20,21,22,23,26
24,25,28,29,30,31
35,36,34,33,27,32
=================

notes:
1) in N=1 we start (and end!) with n=2
2) case N=2 has the only solution, while
3) cases N>2 allow also other (any?) initial n's.