Prime Numbers Magic Squares

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CONTENTS

A Large order-3

This magic square consists of 9 consecutive, 93-digit prime numbers.

Minimum consecutive primes -3

This order-3 uses consecutive primes not in arithmetic progression.

Minimum consecutive primes -4

This order-4 has a magic sum of 258

Minimum consecutive primes -5

This order-5 has a magic sum of 1703. But now one with S = 313

Minimum consecutive primes -6

An order-6 pandiagonal magic square with a surprisingly small sum.

A Small order-3

This is the smallest possible with primes in arithmetic progression.

Primes in arithmetic progression

An order-4 pandiagonal magic square using 14 or 15 digit primes.

Orders 3 & 8 use consecutive primes

73 consecutive primes from 3 to 373 together form 2 magic squares.

Orders 4, 5, 6 use consecutive primes

Prime # 37 to 103, 107 to 239 and 241 to 457 make 3 magic squares.

A Bordered prime magic square

Orders 8, 6 and 4 using distinct 4-digit primes.

Order-3 with smallest sum

These primes are neither consecutive or in arithmetical progression.

Two palprime magic squares

All numbers in these order-3's are 11-digit palindromic primes.

Order-13 constant difference

Nested squares of orders 13, 11, 9, 7, 5, 3, 1.

Order-7 two way prime pandiagonal

Even when the unit digit of each number is removed.

Two minimum difference squares

Order-5 add and multiply squares have minimum differences.

Prime number - Smith number

Two order-3 magic squares, sums are 822 and 411.

Anti-magic squares have prime sums

Two order-3 squares with minimal solutions.

Orthomagic squares of squares

Squares of primes form a square with rows and columns magic.

Primes and composites

The prime numbers form a capitol T in this order-5 magic square.

Order-5 ...... with NO primes

25 consecutive composite numbers make up this super-magic square.

Order-11 Prime-magical square

This array contains 24 different reversible 11-digit primes.

Previously posted prime squares

Links to other prime magic squares previously posted on this site.

A Large order-3

The following 93 digit number is the first of ten consecutive primes in arithmetic progression. Each one is 210 larger then the previous one.

100 99697 24697 14247 63778 66555 87969 84032 95093 24689 19004 18036 03417 75890 43417 03348 88215 90672 29719.

 
p + 1680 p + 210 p + 1260
p + 630 p + 1050 p + 1470
p + 840 p + 1890 p + 420
This series was discovered in March, 1998 by Manfred Toplic of Austria.

An order-3 prime number magic square may be constructed using the first 9 or the last nine of these primes.
This magic square uses the last nine. To save space, p is used to represent this large number in each cell. The magic constant then is 3p + 3150.

A smaller order-3 consecutive primes magic square could be constructed with the nine prime series starting with 99 67943 20667 01086 48449 06536 95853 56163 89823 64080 99161 83957 74048 58552 90714 75461 11479 96776 94651.
This series also has a difference of 210 between successive primes.

Minimum consecutive primes -3

1480028201 1480028129 1480028183
1480028153 1480028171 1480028189
1480028159 1480028213 1480028141
 These are the only two 3 x 3 magic squares composed of consecutive primes under 231. In each case the series consists of 3 triplets with a starting difference of 6 and an internal difference of 12.

 

Both were found by Harry Nelson who found 18 other magic squares of this type, the highest sequence starting with 9 55154 49037. All are greater then 231 which is 21474 83648.

H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p.214

1850590129 1850590057 1850590111
1850590081 1850590099 1850590117
1850590087 1850590141 1850590069
Type 1
P8 P1 P6
P3 P5 P7
P4 P9 P2

Theoretically, there are two different types of arrays possible. Both of the above magic squares are type 1. There are no type 2 consecutive prime magic squares under 231, and it is not known if any even exist.
Addendum: August 4, 1999
Harry J. Smith confirms that Aale de Winkel has discovered a Type 2 magic square!

Type 1 is the only magic square possible using consecutive (prime & composite) numbers.

In each case, in these 2 squares, the numbers in the cells indicate the magnitude (order) of the number in the series of 9 numbers.

See my Type 2 Order-3 page.

From a letter by Harry J. Smith of Saratoga, CA, to Dr. Michael W. Ecker dated Dec. 8/90. Farrago IX disk 4

Type 2
P8 P1 P7
P4 P5 P6
P3 P9 P2

Minimum consecutive primes -4

37

83

97

41

53

61

71

73

89

67

59

43

79

47

31

101

The primes 31 to 101 form a magic square with a magic sum of 258.
Author Allan W. Johnson, Jr. shows another order-4 using primes 37 to 103 and magic sum 276.
These primes are not in arithmetic progression.

This is in answer to problem 962 originally posed by Frank Rubin.

Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp.152-153

Minimum consecutive primes -5

281

409

311

419

283

359

379

349

347

269

313

307

389

293

401

397

331

337

271

367

353

277

317

373

383

The primes 269 to 419 form a magic square with a magic sum of 1703.
Author Allan W. Johnson, Jr. shows another order-5 using smaller primes 181 to 389 but a magic sum 1704.

This also in answer to problem 962 originally posed by Frank Rubin.

Journal of Recreational Mathematics, vol. 14:2, 1981-82, pp.152-153

59 107 71 23 53
13 37 113 61 89
43 41 83 79 67
101 19 17 103 73
97 101 29 47 31
Addendum  September 2009

Max Alekseyey advised me that the above is not the smallest possible order-5 prime simple magic square. Several smaller ones are shown at. [1]. This is the smallest, with S = 313.

Also shown at that site is a simple order-6 magic square with S = 484

[1] http://digilander.libero.it/ice00/magic/prime/orderConstant.html

Minimum consecutive primes -6

67 193 71 251 109 239
139 233 113 181 157 107
241 97 191 89 163 149
73 167 131 229 151 179
199 103 227 101 127 173
211 137 197 79 223 83
This pandiagonal magic square consists of the thirty-six consecutive primes from 67 to 251. This is the smallest series of primes possible for forming a pandiagonal order-6 magic square. See [1] (above ) for a simple order-6 with S = 484.
There are 24 different combinations of numbers that equal the magic sum of 930. The 6 rows, 6 columns, 2 main diagonals, and 10 pan diagonal pairs.

The author also shows two order-6 pandiagonal magic squares with smaller series of primes. These both use 36 primes from the series 3 to 167.

A. W. Johnson, Jr. Journal of Recreational Mathematics, vol. 23:3, 1991, pp.190-191

A Small order-3

 
1669 199 1249
619 1039 1459
829 1879 409
This order-3 magic square is the smallest possible with primes in arithmetic progression   (but not consecutive).

David Wells, Penguin Dictionary of Curious & Interesting Numbers, 1986.

This magic square was first published by Dudeney in 1917.

H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, p. 246

Primes in arithmetic progression

39,064,930,015,753

98,983,213,040,353

66,719,522,180,953

89,765,015,651,953

103,592,311,734,553

52,892,226,098,353

75,937,719,569,353

62,110,423,486,753

80,546,818,263,553

57,501,324,792,553

108,201,410,428,753

48,283,127,404,153

71,328,620,875,153

85,155,916,957,753

43,674,028,709,953

94,374,114,346,153

This magic square is pandiagonal with the magic sum of 294,532,680,889,012.
As with all order-4 pandiagonal magic squares, the following all sum correctly:

  • 4 rows
  • 4 columns
  • 8 diagonals
  • 16                 2 x 2 squares (including wrap-around) This qualifies it as a most-perfect magic square.
  • 16 corners of 3 x 3 squares
  • 16 corners of 4 x 4 squares

It is composed of the top16 of 22 prime numbers in arithmetic progression, and a common difference of 4,609,098,694,200.

The smallest possible order-4 magic square of this type may be made from the series starting with 53,297,929 and a common difference of 9,699,690.

The longest known arithmetic progression, all of whose members are prime numbers, contains 22 terms. The first term is 11,410,337,850,553 and the common difference is 4,609,098,694,200.
It was discovered on 17 March 1993 at Griffith University, Queensland.

An arithmetic progression is a sequence of numbers where each is the same amount more than the one before. For example, 5, 11, 17, 23 and 29. All of these are prime numbers, the first term is 5 and the common difference is 6.
In this example, the primes are not consecutive, because the 7, 13 and 19 are missing.

Orders 3 & 8 use consecutive primes

3 367 97 5 281 263 173 271
137 19 151 179 269 347 257 101
359 239 373 41 227 61 71 89
31 313 349 353 107 167 127 13
241 113 29 193 59 283 211 331
197 53 191 307 163 83 317 149
311 199 47 131 17 233 293 229
181 157 223 251 337 23 11 277
109 7 103
67 73 79
43 139 37
This pair of magic squares are constructed using the 73 consecutive primes from 3 to 373.

 

73 is a prime number, as is 11, the sum of the two orders.

Gakuho Abe, Journal of Recreational Mathematics, 10:2, 1977-78, pp. 96-97

Orders 4, 5, 6 use consecutive primes

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

Order 4 Uses the consecutive primes from 37 to 103

Together these three magic squares use the 77 consecutive prime numbers from 37 to 457.

 

 

A. W. Johnson, Jr., Journal of Recreational Mathematics, vol. 15:1,1982-83, pp.17-18

107 229 181 239 109
233 131 191 137 173
149 139 223 127 227
179 199 113 211 163
197 167 157 151 193

Order 5 Uses the consecutive primes from 107 to 239

251 389 311 449 347 353
313 359 293 373 379 383
397 271 419 263 401 349
269 317 367 421 283 443
439 307 277 337 409 331
431 457 433 257 281 241

Order 6 Uses the consecutive primes from 241 to 457

A Bordered prime magic square

2621 2477 2039 1289 3251 1583 3533 2207
3257 1361 3491 2393 2333 2963 1709 1493
2609 1811 2837 2087 2687 1889 2939 2141
2777 2819 2753 1823 1223 3701 1931 1973
2351 2879 1049 3527 2927 1997 1871 2399
1283 2339 2861 2063 2663 1913 2411 3467
1559 3041 1259 2357 2417 1787 3389 3191
2543 2273 2711 3461 1499 3167 1217 2129
This order-8 magic square borders a pandiagonal order-6 magic square, which borders an associated order-4 magic square.
All integers are distinct 4 digit prime numbers.

 

 

A. W. Johnson, Jr., Journal of Recreational Mathematics 15:2, 1982-83, p. 84

Order-3 with smallest sum

 
43 1 67
61 37 13
7 73 31
The constant of this (upper) magic square is 111.

In 1913, Dudeney listed the first solvers of prime magic squares of orders 3 to 12.
This one is by himself.
However, for orders 3 and 12 (and presumably others) the number 1 was used. By present day convention, the number 1 is no longer permitted in prime number magic squares.

The constant of this (lower) magic square is 177.

Note that the primes in these magic squares are neither consecutive nor in arithmetic progression.
This magic square consists of 3 triplets with starting differences of 42, and internal differences of 12.

H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, p. 123

 
101 5 71
29 59 89
47 113 17

Two palprime magic squares

10797779701 14336063341 12568586521
14338283341 12567476521 10796669701
12566366521 10798889701 14337173341
10915551901 12133533121 11527872511
12137973121 11525652511 10913331901
11523432511 10917771901 12135753121

These beautiful magic squares, consisting of 11-digit palindromic primes, are by Carlos Rivera and Jaime Ayala.
As with all order-3 magic squares, these contain 3 triplets. In the case of the first magic square, the triplets start with 10796669701, 10797779701, and 10798889701 for a common difference of 1110000. The common difference within each triplet is 1769696820.

I received the first one on May 22, 1999 by e-mail. The second magic square arrived two days later. Thanks Carlos & Jaime.

Their Prime Puzzles and Problems page is at http://www.primepuzzles.net

Order-13 constant difference

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

This 13 x 13 magic square of all prime numbers contains an 11 x 11, 9 x 9 7 x 7, 5 x 5, 3 x 3 magic squares.
The magic constants of the respective squares are 70681, 59807, 48933, 27185, 16311.
The common difference between each of these constants is 10874, including the difference between the 3 x 3 square and the center number 5437.

Both this and the next magic square were composed by a hobbyist while serving time in prison.

This is a concentric (not bordered) magic square.

J. S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons, 1966, pp92 94.

Order-7 two way prime pandiagonal square

11

3851

9257

1747

6481

881

5399

6397

827

5501

71

3779

9221

1831

3881

9281

1759

6361

911

5417

17

839

5381

101

3797

9227

1861

6421

9311

1777

6367

941

5441

29

3761

5387

131

3821

9239

1741

6451

857

1801

6379

821

5471

47

3767

9341

The magic sum for this square is 27627 for every row, column, main diagonal and broken diagonal pair.

If a new square is constructed by removing the units digit from each number (11 becomes 1, 3851 becomes 385, etc), it will have the magic sum of 2760 for every row, column diagonal and broken diagonal pair!

J. S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons, 1966, pp92 94.

Two minimum difference squares

Add
41 79 17 13 61
53 3 83 67 7
59 97 5 23 29
11 31 37 89 43
47 2 71 19 73

 

These two squares each contain the 25 primes that are less then 100.

 

Add: The maximum sum of any row, column or diagonal is 213

The minimum sum is 211

The difference (which is the minimum possible) is 2

Multiply
17 59 71 89 3
31 79 5 37 41
23 2 67 73 83
29 47 13 11 97
53 43 61 7 19
Multiply: The maximum product of any line, column or diagonal is 19,013,871

The minimum product is 18,489,527

The difference which is also the minimum possible, is 524,344

 

Journal of Recreational Mathematics vol.26:4, 1994, pp308,309
Solutions by Michael Reid to puzzle 2094 originally posed by Rodolfo Kurchan

Prime number - Smith number

 
94 382 346
526 274 22
202 166 454

A. Smith Numbers

A Smith number has the following property.

The sum of its digits is equal to the sum of the digits of its prime factors.
Take 526 as an example. The sum of 5+2+6=13 and the sum of the digits of its prime factors (2 and 263) also equals 13.

There are an infinite amount of Smith numbers, 81 within the natural numbers 1 to 2000. 29,928 among the first 1,000,000 integers.
For every repunit number whose prime factors are known, a Smith number can be constructed.

Square B is formed by dividing each number in A by 2.

The constant of magic square A is 822 (not a Smith number),
and the constant of magic square B is 411 (not a prime).

 

Martin Gardner, Penrose Tiles to Trapdoor Ciphers, 1989, pp 299-301
See also
David Well, Curious and Interesting Numbers, Penguin, 1986, p 187 (# 4,937,775)

 
47 191 173
263 137 11
101 83 227

B. Prime Numbers

Anti-magic squares have prime sums

  1 2 4 7
  16 3 6 25
  12 8 7 27
19 29 13 17 11
A normal antimagic square is an n x n array of integers from 1 to n2, arranged so that the rows, columns and diagonals sum to different but consecutive numbers. There are no order-2 or 3 normal antimagic squares.

Here we relax the definition to use non-consecutive, non-distinct numbers and show two order-3 squares that involve prime number sums

A. Every sum has only 1 prime factor.

B. The sums are the first eight primes

The squares are by Torben Mogensen and appeared on an Internet newsgroup Aug. 14, 1997.
The antimagic definition is by J. A. Lindon and appeared in J. S. Madachy, Mathematics On Vacation, Nelson, 1966, p. 103.

  2 0 1 3
  5 0 6 11
  12 5 0 17
13 19 5 7 2

Orthomagic squares of squares

 
112 232 712
612 412 172
432 592 192
In a new approach to searching for order-3 magic squares consisting of all perfect squares, Kevin Brown has investigated squares which have the rows and columns summing the same , but not the diagonals. He calls these orthomagic squares of squares, of OMSOS for short.

He found 91 primitive OMSOS squares with common sum less then 30,000; and proved that this type of square can not have the diagonals summing correctly. Of the 91 primitive squares, 56 have a common sum that is a perfect square.

Interestingly, he found that three of the other 35 squares consist of all prime numbers. Here is the smallest one.
The common sum of the rows and columns is 5691

See his paper on OMSOS here.

Primes and composites

 
19 23 11 5 7
1 10 17 24 13
22 14 3 6 20
8 16 25 12 4
15 2 9 18 21
The prime numbers in this pandiagonal magic square form a capitol T.

It was constructed by Dr. C. Planck and published in 1917.

As was common in that era, the one was included as a prime number.

By convention, the number 1 is no longer permitted in prime magic squares..

H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, p. 246

Order-5 ...... with NO primes

 

1328

1342

1351

1335

1344

1350

1334

1343

1332

1341

1347

1331

1340

1349

1333

1339

1348

1337

1346

1330

1336

1345

1329

1338

1352

This magic square consists of 25 consecutive composite numbers. It is the smallest possible such magic square of order-5.
It is a pandiagonal associative, complete and self-similar magic square with a magic sum of 6700.

Including the usual 5 rows, 5 columns and 10 diagonals, there are 328 different ways to form the sum of 6700 using 5 numbers.

Refer to "A Deluxe Magic Square" on my Pandiag.htm page for a full discussion, including definitions, of this type of magic square.

You could make an order-25 composite magic square like the above using the 625 consecutive numbers starting with  11,000,001,446,613,354.

See David Wells, Curious and Interesting Numbers, Penguin, 1986, p. 195.

Order-11 Prime-magical square

3 7 9 7 9 9 1 3 9 7 3
7 9 1 9 1 9 1 7 9 9 9
7 1 1 9 1 9 3 9 7 9 9
1 1 1 1 3 7 9 9 7 7 1
1 1 1 7 1 7 1 9 3 3 1
1 7 3 7 1 7 9 3 7 1 1
1 7 9 9 1 3 1 1 3 3 3
3 9 1 9 1 9 1 1 3 3 7
7 7 9 9 7 1 1 3 7 9 1
7 9 3 3 3 7 7 7 7 3 9
3 3 9 3 3 9 1 3 9 1 3
This 11 x 11 square is not magic in the usual sense. The rows, columns and diagonals do not add up to the same constant.
In this case, the rows, columns and diagonals are distinct, reversible and non-palindromic primes.

So this square consists of 48 different 11-digit primes!

The puzzle was designed by Carlos Rivera and his friend Jaime Ayala and posted on their excellent Prime Puzzles and Problems page about a year ago (June, 1998).

 

 

See much more on this subject as well as lots more on prime numbers at http://www.primepuzzles.net

The above solution was sent to Carlos June 6, 1999 by Jurgen T. W. A. Baumann.

Previously posted prime squares

The following are prime magic squares that were previously posted to this site.

For convenience, I list them here with links to the corresponding pages.

Consecutive Prime Numbers Order-9 magic square ----- Material From REC
                  This order-9 magic square is composed of the 81 consecutive prime numbers 43 to 491.

Order-16 Prime Number Magic Square ---------------- Material From REC
                  This magic square contains inlays of each even order magic square from 4 to 14.

Prime Number heterosquares --------------------------- Unusual Magic Squares
                   Two order-3 heterosquares by Carlos Rivera. All numbers are prime.

Orders 4 & 5 Perfect Prime Squares ------------------- Prime Number Patterns
                   All rows, columns and the two main diagonals are distinct prime numbers when read in either direction.

Order-6 Perfect Prime Squares ------------------------ Prime Number Patterns
                  Rivera and  Ayala's two order-6 squares which each contain twenty-eight 6 digit primes.

Order-3 Super-Perfect Prime Square ------------------ Prime Number Patterns
                  1 of the 24 possible order-3 perfect prime squares. The partial diagonal pairs are also prime numbers

Type 2 - Order-3 Minimum consecutive primes -------- Type 2 Order-3
                  Discusses Type 2 m.s. and shows the two smallest consecutive primes order-3 magic squares. Aug. 8/99

 

This page was originally posted June 1999
It was last updated July 21, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz