Problems & Puzzles: Puzzles

Puzzle 3.- Magic Squares with consecutive primes

Harry Nelson was the first in produce (and he won a $100.00 prize offered by Martin Gardner) a 3x3 matrix containing only consecutive primes.

His solution has in the central cell the prime 1480028171. The other cells has the following primes +/-12 ; +/-18, +/-30, +/-42.

(Ref. 2, p. 18)

While it was not easy produce the solutions, for sure you can assign the primes obtained by Nelson. Do you want to try ?

We know that Nelson got more than other 20 such magical squares.

1.- Is there any method for doing that ?

2.- Is there any 4x4 matrix with consecutive primes ?

Solution

Aale de Winkel sent (30/06/99) the following comment: :

"...given a regular order n magic square (numbers 1..n^2 : magic number t=n(n^2+1)/2) substitute each number (i) with a number (x + i * d) a magic square is obtained     with magic number nx+td = n(x+n^2+1)/2. hence any n^2 primes in arithmetic progression thus gives magic squares of order n."

According to this method you can get two 3x3 magic squares using the 10 consecutive primes in A.P. found by Dubner, Tony Forbes, Manfred Toplic et al.

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Several 4x4 prime magic squares can be found at: http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.4x4.prime.html
 

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Luke Pebody sent the following solutions to Q2 (April 2005):

31-101
67 89 43 59
47 79 101 31
83 37 41 97
61 53 73 71

37-103
79 59 71 67
53 73 61 89
97 101 41 37
47 43 103 83

79 67 83 47
41 101 61 73
59 71 43 103
97 37 89 53

1229-1321
1289 1279 1291 1259
1231 1283 1301 1303
1277 1237 1297 1307
1321 1319 1229 1249

1279 1289 1259 1291
1283 1303 1231 1301
1237 1277 1307 1297
1319 1249 1321 1229

1301 1231 1303 1283
1259 1291 1279 1289
1321 1319 1229 1249
1237 1277 1307 1297

4931-5021
4951 4931 5021 4993
4999 4967 4973 4957
5003 4987 4969 4937
4943 5011 4933 5009

12553-12689
12653 12577 12589 12659
12583 12611 12647 12637
12689 12619 12601 12569
12553 12671 12641 12613

3259909-3260063
3260051 3260017 3259933 3259979
3259999 3259931 3260063 3259987
3260021 3260029 3259957 3259973
3259909 3260003 3260027 3260041

3324329-3324521
3324389 3324521 3324407 3324361
3324353 3324457 3324509 3324359
3324499 3324371 3324341 3324467
3324437 3324329 3324421 3324491

26025107-26025281
26025211 26025239 26025127 26025257
26025203 26025179 26025199 26025253
26025271 26025229 26025227 26025107
26025149 26025187 26025281 26025217

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On October, 2005, Luke Pebody wrote:

Here is a 5x5 magic square

067 047 103 017 079
101 023 053 029 107
059 113 043 061 037
013 089 083 109 019
073 041 031 097 071
 

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Anurag Sahay wrote (July 07):

Some more 5x5 examples:

magic sum=313

19 29 113 43 109
103 83 13 67 47
53 71 59 107 23
101 41 31 79 61
37 89 97 17 73


103 59 67 13 71
47 17 89 53 107
31 79 73 101 29
113 97 43 37 23
19 61 41 109 83

79 167 193 107 157
181 151 191 83 97
131 109 127 137 199
139 163 103 197 101
173 113 89 179 149
 

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