# A huge magic square

A magic square is a square matrix of pairwise different natural numbers such that the sum of the entries in each row, each column and each diagonal is the same. Such a square is particularly nice if all its entries are prime numbers. For instance, if the numbers a + kr for k=1,2,...,8,9 are prime, the square

a+4r a+9r a+2r

a+3r a+5r a+7r

a+8r a+r a+6r

is magic. These numbers are prime e.g. for r=210 and a=-11 or a=199. In 1961 a magic square with 169 different primes which formed no arithmetic sequence was published. It was the work of an anonymous inmate of one of the American prisons. Here is the square:

1153 8923 1093 9127 1327 9277 1063 9133 9661 1693 991 8887 8353

9967 8161 3253 2857 6823 2143 4447 8821 8713 8317 3001 3271 907

1831 8167 4093 7561 3631 3457 7573 3907 7411 3967 7333 2707 9043

9907 7687 7237 6367 4597 4723 6577 4513 4831 6451 3637 3187 967

1723 7753 2347 4603 5527 4993 5641 6073 4951 6271 8527 3121 9151

9421 2293 6763 4663 4657 9007 1861 5443 6217 6211 4111 8581 1453

2011 2683 6871 6547 5227 1873 5437 9001 5647 4327 4003 8191 8863

9403 8761 3877 4783 5851 5431 9013 1867 5023 6091 6997 2113 1471

1531 2137 7177 6673 5923 5881 5233 4801 5347 4201 3697 8737 9343

9643 2251 7027 4423 6277 6151 4297 6361 6043 4507 3847 8623 1231

1783 2311 3541 3313 7243 7417 3301 6967 3463 6907 6781 8563 9091

9787 7603 7621 8017 4051 8731 6427 2053 2161 2557 7873 2713 1087

2521 1951 9781 1747 9547 1597 9811 1741 1213 9181 9883 1987 9721

It has a very interesting property. If you delete the extreme rows and columns, you get a magic square again with 121 entries. Deleting again the extreme rows and columns you get a magic square with 100 entries, and so on. Incindentally, the largest increasing arithmetical sequence of prime numbers known up to date has 17 elements. It was found by the American mathematician Sol Weintraub in 1977 (with the use of a computer, of course) and its elements are

ak=3,430,751,869 + 87,297,210 k,

for k=0,1,..., 16.

Andrzej Makowski