Find the array of single digits which contains the maximum possible number of primes, where allowable primes may lie along any horizontal, vertical, or diagonal line. For , 11 primes are maximal and are contained in the two distinct arrays

giving the primes (3, 7, 13, 17, 31, 37, 41, 43, 47, 71, 73) and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97), respectively. For the array, 18 primes are maximal and are contained in the arrays

The best array is

which contains 30 primes: 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 97, 113, 157, 179, ... (Sloane's A032529). This array was found by Rivera and Ayala and shown by Weisstein in May 1999 to be maximal and unique (modulo reflection and rotation).

The best arrays known are

all of which contain 63 primes. The first was found by C. Rivera and J. Ayala in 1998, and the other three by James Bonfield on April 13, 1999.

The best prime arrays known are

each of which contains 116 primes. The first was found by C. Rivera and J. Ayala in 1998, and the second by Wilfred Whiteside on April 17, 1999.

The best prime arrays known are

each of which contain 187 primes. One was found by S. C. Root, and the others by M. Oswald in 1998.

The best prime array known is

which contains 281 primes and was found by Wilfred Whiteside on April 29, 1999.

The best prime array known is

which contains 382 primes and was found by Wilfred Whiteside On Oct. 31, 1999.

Heuristic arguments by Rivera and Ayala suggest that the maximum possible number of primes in , , and arrays are 58-63, 112-121, and 205-218, respectively.

Array, Prime Arithmetic Progression, Prime Constellation, Prime Magic Square, Prime String