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. Dudeney (1970) records its discovery in 1900 (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).

L. Sallows (pers. comm., Nov. 14, 2013) gave the prime magic square

which has the same magic constant 120 as Johnson's example but a largest entry of 71 instead of 73.

Gardner (1984, pp. 86-87) attributes to J. N. Muncey a 1913 proof that the smallest nontrivial 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.