CONTENTS

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

This order3 uses consecutive primes not in arithmetic
progression. 

This order4 has a magic sum of 258 

This order5 has a magic sum of 1703. But now one with S = 313 

An order6 pandiagonal magic square with a surprisingly small
sum. 

This is the smallest possible with primes in arithmetic
progression. 

An order4 pandiagonal magic square using 14 or 15 digit primes. 

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

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

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

These primes are neither consecutive or in arithmetical
progression. 

All numbers in these order3's are 11digit palindromic primes. 

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

Even when the unit digit of each number is removed. 

Order5 add and multiply squares have minimum differences. 

Two order3 magic squares, sums are 822 and 411. 

Two order3 squares with minimal solutions. 

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

The prime numbers form a capitol T in this order5 magic square. 

25 consecutive composite numbers make up this supermagic
square. 

This array contains 24 different reversible 11digit primes. 

Links to other prime magic squares previously posted on this
site. 
A Large order3
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 order3 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 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.
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 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 
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 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
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 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 
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 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 
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
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 
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
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 
A Small order3
1669 
199 
1249 
619 
1039 
1459 
829 
1879 
409 

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 
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 order4 pandiagonal magic squares, the following all sum correctly:
 4 rows
 4 columns
 8 diagonals
 16 2 x 2 squares (including wraparound) This qualifies it
as a mostperfect 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 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.
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, 197778, pp. 9697 
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. 15:1,198283, pp.1718 
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 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 
Order3 with smallest sum

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 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
Order13 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. 
Order7 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 299301
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 
Antimagic 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 n^{2}, arranged so that the
rows, columns and diagonals sum to different but consecutive
numbers. There are no order2 or 3 normal antimagic squares.
Here we relax the definition to use nonconsecutive, nondistinct
numbers and show two order3 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
11^{2} 
23^{2} 
71^{2} 
61^{2} 
41^{2} 
17^{2} 
43^{2} 
59^{2} 
19^{2} 

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.
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 
Order5 ...... 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 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.
Order11 Primemagical 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 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. 
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 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
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
