Prime Numbers Magic Squares

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