Lee Sallows
Home > Magic Squares > Minimal 4x4 of primes

The following is an email sent to Harvey Heinz on August 11 2007. I don't see the square of primes mentioned below on his site http://www.magic-squares.net/.


Dear Harvey,
                     Browsing your excellent website I note that among the various 4×4 squares using prime numbers you give, none of them is assumed to be minimal, in the sense of using the smallest set of 16 distinct (not necessarily consecutive) primes.

This led me to look at http://mathworld.wolfram.com/PrimeMagicSquare.html where a square due to Allan Wm Johnson having a magic sum of 120 is shown. It is the closest approach to a minimal square that I have found on the web.

Using a computer program able to generate every 4×4 magic square formable from a given set of 16 integers, I examined every possible set of 16 among the first 20 or so primes. My results confirm that Johnson's square has the lowest magic sum of all: 120. However, there exists another square with a magic sum of 120 in which the largest prime appearing is 71 rather than Johnson's 73. It would seem fair therefore to describe this square as the minimal case:

           Johnson's:                         new square:

 3  61  19  37     17   3  53  47
43  31   5  41      7  71  31  11
 7  11  73  29     59   5  13  43
67  17  23  13     37  41  23  19

On the other hand, if 1 is acceptable within a square of primes, the minimal case is as follows:

23   1  31  47
 5  37  41  19
71  11  13   7
 3  53  17  29

The magic sum is 102.

Regards, Lee

ps
Here is something unusual:

 -6 -16  18 -21
-36  12  -4   3
  8  -9 -24   0
  9 -12 -15  -7

It is a 4×4 square (including negative numbers) with a magic sum of -25. What makes it special is that the 3×3 square embedded in the top left hand corner is a multiplicative magic square with magic product 1728.
Last Updated:
14-6-2020
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