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. |
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.
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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. |
A smallest 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.
H.H. 1999
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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 |
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Type 1
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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 |
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Type 2
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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 |
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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 |
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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 |
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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 |
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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 |
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:
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.
H.H. 1999
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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 |
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. 15:1,1982-83, pp.17-18 |
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Order 5 Uses the consecutive primes from 107 to 239 |
Order 6 Uses the consecutive primes from 241 to 457 |
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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 |
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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 |
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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
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.
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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
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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
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Multiply
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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 299-301 |
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B. Prime Numbers |
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A. Every sum has only 1 prime factor. 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 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 B. The sums are the first eight primes
The squares are by Torben Mogensen and appeared on an Internet newsgroup
Aug. 14, 1997. |
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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. See his paper on OMSOS here. |
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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 |
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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.
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 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. |
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
3 Digits All Prime ------------------------------------ Prime Number Patterns
3 semi-magic squares of 3-digit primes
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
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Last updated
September 24, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 1998, 1999, 2000 by Harvey D. Heinz