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Prime Array

Find the $m\times n$ 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 $m=n=2$, 11 Primes are maximal and are contained in the two distinct arrays

\begin{displaymath}
A(2,2)=\left[{\matrix{1 & 3\cr 4 & 7\cr}}\right], \left[{\matrix{1 & 3\cr 7 & 9\cr}}\right],
\end{displaymath}

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 $3\times 2$ array, 18 Primes are maximal and are contained in the arrays
$\displaystyle A(3,2)$ $\textstyle =$ $\displaystyle \left[\begin{array}{ccc}1 & 1 & 3\\  9 & 7 & 4\end{array}\right],...
...rray}\right], \left[\begin{array}{ccc}1 & 7 & 2\\  4 & 3 & 9\end{array}\right],$  
  $\textstyle \phantom{=}$ $\displaystyle \left[\begin{array}{ccc}1 & 7 & 5\\  4 & 3 & 9\end{array}\right],...
...rray}\right], \left[\begin{array}{ccc}1 & 7 & 9\\  4 & 3 & 2\end{array}\right],$  
  $\textstyle \phantom{=}$ $\displaystyle \left[\begin{array}{ccc}1 & 7 & 9\\  4 & 3 & 4\end{array}\right],...
...rray}\right], \left[\begin{array}{ccc}3 & 7 & 6\\  4 & 1 & 9\end{array}\right].$  

The best $3\times 3$, $4\times 4$, and $5\times 5$ prime arrays known were found by C. Rivera and J. Ayala in 1998. They are

\begin{displaymath}
A(3,3)=\left[{\matrix{
1 & 1 & 3\cr
7 & 5 & 4\cr
9 & 3 & 7\cr}}\right],
\end{displaymath}

which contains 30 Primes,

\begin{displaymath}
A(4,4)=\left[{\matrix{
1 & 1 & 3 & 9\cr
6 & 4 & 5 & 1\cr
7 & 3 & 9 & 7\cr
3 & 9 & 2 & 9\cr}}\right],
\end{displaymath}

which contains 63 Primes, and

\begin{displaymath}
A(5,5)=\left[{\matrix{
1 & 1 & 9 & 3 & 3\cr
9 & 9 & 5 & 6 & ...
...& 1 & 7\cr
3 & 3 & 7 & 3 & 1\cr
3 & 2 & 9 & 3 & 9\cr}}\right],
\end{displaymath}

which contains 116 Primes. S. C. Root found the a $6\times 6$ array containing 187 primes:

\begin{displaymath}
A(6,6)=\left[{\matrix{
3 & 1 & 7 & 3 & 3 & 3\cr
9 & 9 & 5 & ...
...cr
3 & 4 & 9 & 1 & 9 & 9\cr
3 & 7 & 9 & 3 & 7 & 9\cr}}\right].
\end{displaymath}

In 1998, M. Oswald found five new $6\times 6$ arrays with 187 primes:
$\quad\left[{\matrix{1 & 3 & 9 & 1 & 9 & 9\cr 3 & 1 & 7 & 2 & 3 & 4\cr 9 & 9 & 4...
...& 5 & 7 & 1 & 3\cr 9 & 8 & 3 & 6 & 1 & 7\cr 9 & 1 & 7 & 3 & 3 & 3\cr}}\right], $
$\quad\left[{\matrix{3 & 1 & 7 & 3 & 3 & 3\cr 9 & 9 & 5 & 6 & 3 & 9\cr 1 & 1 & 8...
...& 6 & 3 & 7 & 3\cr 3 & 4 & 9 & 1 & 9 & 9\cr 9 & 7 & 9 & 3 & 7 & 9\cr}}\right], $
$\quad\left[{\matrix{3 & 1 & 7 & 3 & 3 & 3\cr 9 & 9 & 5 & 6 & 3 & 9\cr 1 & 1 & 8...
... & 6 & 3 & 7 & 3\cr 3 & 4 & 9 & 1 & 9 & 9\cr 9 & 9 & 9 & 2 & 3 & 3\cr}}\right].$


Rivera and Ayala conjectured and Weisstein demonstrated by direct computation in May 1999 that the 30-prime solution for $A(3,3)$ is maximal and unique. Heuristic arguments by Rivera and Ayala suggest that the maximum possible number of primes in $4\times 4$, $5\times 5$, and $6\times 6$ arrays are 58-63, 112-121, and 205-218, respectively.

See also Array, Prime Arithmetic Progression, Prime Constellation, Prime String


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.

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



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© 1996-9 Eric W. Weisstein
1999-05-26