Prime Magic Square

A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the  prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the  magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the  prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The  prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).

According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including the number 1 is of order 12, illustrated above.

The  square whose entries are consecutive primes illustrated above was discovered by Nelson (Guy 1994, p. 18; Rivera) in response to a challenge by Martin Gardner. Nelson collected Gardner's $100 prize, and also found 20 other such squares (Guy 1994, p. 18).

The amazing square above (Madachy 1979, pp. 93-94) is a  prime magic border square, so that the , , ..., and  subsquares are all also prime magic squares.

REFERENCES:

Dewdney, A. K. "Computer Recreations: How to Pan for Primes in Numerical Gravel." Sci. Amer. 259, pp. 120-123, July 1988.

Dudeney, E. Problem 408 in Amusements in Mathematics. New York: Dover, 1970.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Heinz, H. "Prime Magic Squares." https://www.magic-squares.net/primesqr.htm.

Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy's Mathematical Recreations. New York: Dover, pp. 85-113, 1979.

Nelson, H. L. "A Consecutive Prime  Magic Square." J. Recr. Math. 20, 214-216, 1988.

Rivera, C. "Problems & Puzzles: Puzzle 003-Magic Squares with Consecutive Primes." https://www.primepuzzles.net/puzzles/puzz_003.htm.

Rivera, C. "Problems & Puzzles: Puzzle 004-Prime-Magical Squares." https://www.primepuzzles.net/puzzles/puzz_004.htm.

Sloane, N. J. A. Sequences A073502 and A073520 in "The On-Line Encyclopedia of Integer Sequences."