A prime number magic square is a magic square made up of only prime numbers. As with other magic squares, the sum of the numbers in any row, column, or main diagonal is the same. Unlike normal magic squares, this sum is not a constant for each square order; it can be different for each square.
Some early prime number magic squares contain the number 1, which was considered prime at the time.
Order 4 examples: .txt
Orders 5,6,7 examples: .txt
See Suzuki.
See OEIS A164843 for prime magic squares by Andrew Lelechenko, (order 5) and Natalia Makarova, (orders 6 to 14).
See also Ultramagic.
Order 3 examples: .txt
Natalia Makarova has associative squares for orders 7 to 20 at OEIS A188537
See also Ultramagic and Most-perfect.
Order 4 examples, (most-perfect): .txt
Orders 5, 6, 7 examples: .txt
See A179440.
"Pandiagonal Magic Squares of Prime Numbers" | |
has squares for orders 6 to 20. See Zimmermann. |
Orders 6, 7, 8 examples: .txt
See OEIS A257316.
Orders 4, 6, 8 examples: .txt
Natalia Makarova. See OEIS A258082.
Orders 7, 8 examples: .txt
From Journal of Recreational Mathematics 15:2, 1982-83, p. 84
Order 13 examples: .txt
Bogdan has numerous magic squares on his site including concentric prime number squares of orders:
13, 15, 17, 19, 29, 31, 35, 61, 107, 125, 145, 231, 351, 453, 469, 503
Also, an order 259 with all prime numbers except the center 6,999,551, (13 x 53 x 10159).
In 2015, Natalia Makarova made concentric prime magic squares of
orders 5 .. 20. See Makarova.
These are minimal 5, 6, 7, 9, 10, 11, 13, 15, 17, 19 and
other 6, 8, 12, 14, 16, 18, 20.
The magic sums are:
5 6 7 9 10 11 13 15 17 19 ------- ------- ------- ------- -------- -------- -------- -------- -------- -------- 1,255 504 4,487 12,249 4,200 26,521 49,439 74,595 128,197 191,159 6 8 12 14 16 18 20 ------- ------- ------- -------- -------- -------- -------- 630 2,040 8,820 16,170 21,840 35,910 54,600
Orders 5, 6, 7 minimal examples:
Orders 8, 11 examples: .txt
See Boyer. Also, there:
Nicolas Rouanet has bimagic squares of primes for orders 8 to 11, 13, 15 to 25.
Huang Jianchao has bimagic squares of consecutive primes for orders 24 to 28.
A073502 "The smallest magic constant for n X n magic square with prime entries (regarding 1 as a prime)."
https://oeis.org/A073502
A073520 "Smallest magic constant for any n X n magic square made from consecutive primes, or 0 if no such magic square exists."
https://oeis.org/A073520
A164843 "The smallest magic constant of an n X n magic square with distinct prime entries."
https://oeis.org/A164843
A179440 "The smallest magic constant of pan-diagonal magic squares which consist of distinct prime numbers."
https://oeis.org/A179440
A188537 "The smallest constant of an n X n associative magic square composed of distinct primes."
https://oeis.org/A188537
A257316 "Smallest magic constant of ultramagic squares of order n composed of distinct prime numbers."
https://oeis.org/A257316
A258082 "Smallest magic constant of most-perfect magic squares of order 2n composed of distinct prime numbers."
https://oeis.org/A258082
Andrews, W. S. and H. A. Sayles
"MAGIC SQUARES MADE WITH PRIME NUMBERS TO HAVE THE LOWEST POSSIBLE SUMMATIONS."
https://www.jstor.org/stable/27900468?seq=1#metadata_info_tab_contents
Boyer, Christian "Bimagic squares of primes" http://www.multimagie.com/English/BimagicPrimes.htm
Dudeney, Henry E. "MAGIC SQUARES OF PRIMES"
https://archive.org/details/AmusementsInMathematicspdf/page/n182/mode/1up
Golunski, Bogdan "Examples of bordered magic squares"
http://www.number-galaxy.eu/
(See under: mag. squares\magic squares\
magic squares with all prime numbers)
Gutierrez, Eddie N. "ROTATIONAL VARIANTS OF PRIME SQUARES" http://oddwheel.com/primesquare.html
Heinz, Harvey "Prime Numbers Magic Squares" http://www.magic-squares.net/primesqr.htm
Madachy, Joseph. S. "Mathematics on Vacation"
https://epdf.pub/queue/mathematics-on-vacation-also-madachys-mathematical-recreations.html
Makarova, Natalia "Profile"
https://boinc.multi-pool.info/latinsquares/view_profile.php?userid=1
Makarova, Natalia "Concentric magic squares of primes"
http://primesmagicgames.altervista.org/wp/forums/topic/concentric-magic-squares-of-primes/
Sallows, Lee "Minimal 4x4 of primes"
http://www.leesallows.com/index.php?page_menu=Magic%20Squares&pagename=Minimal%204x4%20of%20primes
Sayles, H. A.
"GENERAL NOTES ON THE CONSTRUTION OF MAGIC SQUARES AND CUBES WITH PRIME NUMBERS."
https://www.jstor.org/stable/27900676?seq=2#metadata_info_tab_contents
Suzuki, Mutsumi "Study of Magic Squares."
http://web.archive.org/web/20011122031722/http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.prime.seq.html
Weisstein, Eric W. "Prime Magic Square."
From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/PrimeMagicSquare.html