Magic Squares of Prime Numbers

Description

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.

Magic

Order 4 examples: .txt

I: Ernest Bergholt, C. D. Shuldham. Magic sum 102, (lowest, primes with 1). See Andrews, Sayles.
II: Allan. W. Johnson, Jr. Magic sum 120. See Weisstein.
III: Lee Sallows. Magic sum 120, (lowest). See Sallows.
IV: Allan. W. Johnson, Jr. Magic sum 258. Minimum consecutive primes. See Heinz.
V: André Gérardin. (1925) Magic sum 7,588. Numbers are sums of squares. See Boyer.

Orders 5,6,7 examples: .txt

Order 5: Gakuho Abe. Magic sum 313.
Order 6: Akio Suzuki. Magic sum 484.
Order 7: Akio Suzuki. Magic sum 797.

See Suzuki.

See OEIS A164843 for prime magic squares by Andrew Lelechenko, (order 5) and Natalia Makarova, (orders 6 to 14).

Associative

Pandiagonal

Ultramagic

Orders 6, 7, 8 examples: .txt

Order 6: Max Alekseyev. Magic sum 990.
Order 7: Natalia Makarova. Magic sum 4,613.
Order 8: Natalia Makarova. Magic sum 2,040.

See OEIS A257316.

Most-perfect

Orders 4, 6, 8 examples: .txt

Order 4: Magic sum 240.
Order 6: Magic sum 29,790.
Order 8: Magic sum 24,024.

Natalia Makarova. See OEIS A258082.

Concentric

Orders 7, 8 examples: .txt

Order 7: Allan W. Johnson, Jr. Magic sum 13,853.
Order 8: Allan W. Johnson, Jr. Magic sum 19,000. See Heinz.
The nested order 6 is pandiagonal, the order 4 is associative.

From Journal of Recreational Mathematics 15:2, 1982-83, p. 84

Order 13 examples: .txt

I: By a prison inmate. Magic sum 70,681. See Madachy.
The magic sums for the square and sub-squares are:

70,681 59,807 48,933 38,059 27,185 16,311 5,437 with a common difference of 10,874.

II: Bogdan Golunski. Magic sum 85,111. See Golunski.

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:

Bimagic

Orders 8, 11 examples: .txt

Order 8: Nicolas Rouanet (2018). Magic sums 3,120 and 1,631,000.
Order 11: Christian Boyer (2006). Magic sums 3,497 and 1,578,251.
First known bimagic square of primes.

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.

REFERENCES

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)

Heinz, Harvey "Prime Numbers Magic Squares" http://recmath.org/Magic%20Squares/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

	"Pandiagonal Magic Squares of Prime Numbers"
	has squares for orders 6 to 20. See Zimmermann.