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 ?

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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On November 2012 Natalia Makarova wrote:

I looked puzzle number 3 on your site.
I have participated in this project.
In OEIS has an article:
https://oeis.org/A073520

I made a magic square of order 14 consecutive prime numbers (Magic Constant = 9660):

89	97	193	227	241	521	563	683	911	1063	1231	1237	1283	1321
277	1091	1181	431	1153	757	739	937	941	433	1019	113	139	449
701	283	197	137	827	503	1069	1187	1087	233	499	877	1051	1009
1223	461	541	1259	149	1163	743	719	359	479	647	1291	163	463
311	883	601	239	1289	269	307	967	491	829	211	857	1109	1297
1097	809	1061	1129	823	487	523	587	443	733	199	853	317	599
617	977	167	643	127	709	773	509	577	811	997	971	1361	421
1301	439	1249	419	109	1279	1123	569	383	907	653	251	337	641
659	727	557	691	1327	401	983	179	1151	331	761	1117	157	619
229	991	607	389	257	859	1171	1213	547	1033	1319	631	263	151
1229	191	181	571	863	1307	281	293	1277	379	821	367	887	1013
673	919	929	1303	947	347	397	1039	173	1031	467	313	769	353
593	953	1093	1021	797	787	881	101	103	1049	223	409	457	1193
661	839	1103	1201	751	271	107	677	1217	349	613	373	1367	131

I also was a magic square of order 15, it is in this article OEIS:
https://oeis.org/A164843 ?

And there are a few articles in the drafting of the OEIS magic squares of prime numbers: A173981, A176571, A177434,  A188536,  A188537, A191533, A189188, A191679.

Here is the next larger one (15x15) gotten by Natalia Makarova, published in https://oeis.org/A073520 composed by 225 consecutive primes from 131 to 1619, with a magical sum constant of of 12669.

One more received on December 2012: Stefano Tognon and Natalia Makarova, 35x35 magic square of consecutive primes for 3 to 9941, magic constant 163043, produced on year 2009. Bigger solutions here:http://digilander.libero.it/ice00/magic/prime/orderConstant.html 

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On July 01, 2016, Arkadiusz Wesolowski wrote:

Below are links to sequences giving magic sums of such magic squares of order 3, 4, 5, and 6.
http://oeis.org/A270305
http://oeis.org/A173981
http://oeis.org/A176571
http://oeis.org/A177434

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