
What is the first thing that would you do for begin the analysis of this class of
squares?
Simple: calculate the magic constant for many order of prime squares:
Where P(i) is a function that give us the iexempt prime number.
Look up this table: you see all the possible order for prime squares (remember
that a magic constant is good only if it is an integer):
Order  Magic Constant  Existence 
1  1  YES 
2  8  NO 
3  33  ? 
4  95  NO 
12  4514  ? 
15  9635  ? 
17  14691  ? 
22  34926  ? 
30  97849  NO 
35  162759  ? 
78  2224007  NO 
98  4648955  NO 
124  9912840  ? 
191  39541261  ? 
The table was build using a program that calculate P() and then the magic constant using the above equation.
The max. order analyze is 449, but only until order 254 it was calculated
with two different program, after I use only one program, so are not to
considerate confirmed.
Note that the possible order that have a valid magic constant are very few,
and they are concentrated to the low order.
This is probably due at the fact that at high order we have high difference
between first and latest prime numbers, so the sum grows to a too much
different rhythm.
How we can say about the possible existence of these squares?
If we exclude the order 1 and 2 (because for magic square they are banal),
from the analysis of order and magic constants, we conclude that for order 4,
30, 78, 98 a prime magic square can't exists.
In fact the sum of even odd number can't give odd sum.
The last thing to say about the previous table is that, when I have start to build it, I used a 16Mhz 286, so the max. order analyze was below that 100. The upper order was found with a 266Mhz PII.
Now there's a first result that we can produce using the above table: prove the Muncey theorem.
Theorem:
12 is the smallest order for the prime magic square.
Prof.:
Build all the possible square of order 3, and look that a magic don't exist
(a computer program resolve this test in less that a second).
So, give the Muncey order 12 square to complete the proof.
Magic Square  Tognon Stefano Research 