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 , , and prime arrays known were found by C. Rivera and J. Ayala in 1998. They are

which contains 30 Primes,

which contains 63 Primes, and

which contains 116 Primes. S. C. Root found the a array containing 187 primes:

In 1998, M. Oswald found five new arrays with 187 primes:

Rivera and Ayala conjectured and Weisstein demonstrated by direct computation in May 1999 that the 30-prime solution for is maximal and unique. 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.

**References**

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

Lee, G. ``Winners and Losers.'' *Dragon User.* May 1984.

Lee, G. ``Gordon's Paradoxically Perplexing Primesearch Puzzle.'' http://www.geocities.com/MotorCity/7983/primesearch.html.

Rivera, C. ``Problems & Puzzles (Puzzles): The Gordon Lee Puzzle.'' http://www.sci.net.mx/~crivera/puzzles/puzz_001.htm.

Weisstein, E. W. ``Prime Arrays.'' Mathematica notebook PrimeArray.m.

© 1996-9

1999-05-26