A Large order3 
This magic square consists of 9 consecutive, 93digit prime numbers. 
Minimum consecutive primes 3 
This order3 uses consecutive primes not in arithmetic progression. 
Minimum consecutive primes 4 
This order4 has a magic sum of 258 
Minimum consecutive primes 5 
This order5 has a magic sum of 1703. But now one with S = 313 
Minimum consecutive primes 6 
An order6 pandiagonal magic square with a surprisingly small sum. 
A Small order3 
This is the smallest possible with primes in arithmetic progression. 
Primes in arithmetic progression 
An order4 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 4digit primes. 
Order3 with smallest sum 
These primes are neither consecutive or in arithmetical progression. 
Two palprime magic squares 
All numbers in these order3's are 11digit palindromic primes. 
Order13 constant difference 
Nested squares of orders 13, 11, 9, 7, 5, 3, 1. 
Order7 two way prime pandiagonal 
Even when the unit digit of each number is removed. 
Two minimum difference squares 
Order5 add and multiply squares have minimum differences. 
Prime number  Smith number 
Two order3 magic squares, sums are 822 and 411. 
Antimagic squares have prime sums 
Two order3 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 order5 magic square. 
Order5 ...... with NO primes 
25 consecutive composite numbers make up this supermagic square. 
Order11 Primemagical square 
This array contains 24 different reversible 11digit primes. 
Previously posted prime squares 
Links to other prime magic squares previously posted on this site. 
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.

This series was discovered in March, 1998 by Manfred Toplic
of Austria. An
order3 prime number magic square may be constructed using the first 9 or the last nine of
these primes. 
A smallest order3 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.
H.H. 1999

These are the only two 3 x 3 magic squares composed of consecutive primes under 2^{31}. 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 2^{31} which is 21474 83648. H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p.214 


Type 1

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 2^{31}, 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 Order3 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


The primes 31 to 101 form a magic square with a magic sum of 258. Author Allan W. Johnson, Jr. shows another order4 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, 198182, pp.152153 

The primes 269 to 419 form a magic square with a magic sum of 1703. Author Allan W. Johnson, Jr. shows another order5 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, 198182, pp.152153 

Addendum September 2009 Max Alekseyey advised me that the above is not the smallest possible order5 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 order6 magic square with S = 484 [1] http://digilander.libero.it/ice00/magic/prime/orderConstant.html 

This pandiagonal magic square consists of the thirtysix
consecutive primes from 67 to 251. This is the smallest series of primes
possible for forming a pandiagonal order6 magic square. See [1]
(above ) for a simple order6 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 order6 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.190191 

This order3 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 
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 order4 pandiagonal magic squares, the following all sum correctly:
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 order4 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.
H.H. 1999

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, 197778, pp. 9697 
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 
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. 15:1,198283, pp.1718 

Order 5 Uses the consecutive primes from 107 to 239 
Order 6 Uses the consecutive primes from 241 to 457 

This order8 magic square borders a pandiagonal order6 magic square,
which borders an associated order4 magic square. All integers are distinct 4 digit prime numbers.
A. W. Johnson, Jr., Journal of Recreational Mathematics 15:2, 198283, p. 84 

The constant of this magic square is 111. In 1913, Dudeney
listed the first solvers of prime magic squares of orders 3 to 12. The constant of this magic square is 177. Note that the primes in these magic squares are neither consecutive nor in arithmetic
progression. H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, p. 123 




These beautiful magic squares, consisting of 11digit palindromic primes, are by Carlos
Rivera and Jaime Ayala.
As with all order3 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 email. The second magic square arrived two days later. Thanks Carlos & Jaime.
Their Prime Puzzles and Problems page is at http://www.primepuzzles.net
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.
J. S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons, 1966, pp92 – 94.

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. 
Add

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

Multiply: The maximum product of any line, column or diagonal is
19013871 The minimum product is 18489527 The difference which is also the minimum possible, is 524344
Journal of Recreational Mathematics vol.26:4, 1994, pp308,309 
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. 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. Square B is formed by dividing each number in A by 2. The constant of magic square A is 822 (not a Smith number),
Martin Gardner, Penrose Tiles to Trapdoor Ciphers, 1989, pp 299301 

B. Prime Numbers 

A. Every sum has only 1 prime factor. A normal antimagic square is an n x n array of integers from 1 to n^{2}, arranged so that the rows, columns and diagonals sum to different but consecutive numbers. There are no order2 or 3 antimagic squares. Here we relax the definition to use nonconsecutive, nondistinct numbers and show two order3 squares that involve prime number sums B. The sums are the first eight primes
The squares are by Torben Mogensen and appeared on an Internet newsgroup
Aug. 14, 1997. 



In a new approach to searching for order3 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. See his paper on OMSOS here. 

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 

This magic square consists of 25 consecutive composite numbers. It is the
smallest possible such magic square of order5. It is a pandiagonal associative, complete and selfsimilar 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 order25 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.
H.H. 1999
37979913973 79191917999 71191939799 11113799771 11171719331 17371793711 17991311333 39191911337 77997113791 79333777739 33933913913 
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 nonpalindromic primes. So this square consists of 48 different 11digit 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. 
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 Order9 magic square  Material From REC
This order9 magic square is composed of the 81 consecutive prime numbers 43 to 491.
Order16 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 order3 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.
Order6 Perfect Prime Squares  Prime
Number Patterns
Rivera and Ayala's two order6 squares which each contain twentyeight 6 digit
primes.
Order3 SuperPerfect Prime Square  Prime
Number Patterns
1 of the 24 possible order3 perfect prime squares. The partial diagonal pairs are also
prime numbers
3 Digits – All Prime  Prime Number Patterns
3 semimagic squares of 3digit primes
Type 2  Order3 Minimum consecutive primes  Type 2 Order3
Discusses Type 2 m.s. and shows the two smallest consecutive primes order3 magic squares.
Aug. 8/99
Please send me Feedback about my Web site!
Last updated
September 24, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 1998, 1999, 2000 by Harvey D. Heinz