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Prime magic (Posted on 2018-09-29) Difficulty: 2 of 5
What is the smallest possible constant of a 3×3 magic square containing nine distinct non-composite numbers only?

No Solution Yet Submitted by Ady TZIDON    
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possible solution | Comment 1 of 3

I think this problem relies a bit on the definition of a magic square and the word "small".

If a magic square elements are limited to the numbers 1, 2, 3, ..., n^2, then it is termed a "Normal" Magic Square.  

 

The only solution (constant = 15) to the 3x3 for a Normal Magic Square are 

8 1 6

3 5 7

4 9 2 

 (and its 7 rotations/reflections)

Additional squares may be generated by adding or multiplying the same constant to each term.

Composites, on the other hand, are all positive non-primes excluding 1. Clearly the Normal 3x3 is not made exclusively of non-composites.

 

Then we go for the entire set of 3x3s including the non-normal 3x3s. 

 All 3x3s with distinct integers are of the following form, as per Edouard Lucas:

For integers a,b,c, with 3c being the constant, these are of the form:

(-b+c) (a+b+c) (-a+c)

(-a+b+c) (c) (a-b+c)

(a+c) (-a-b+c) (b+c)


The normal solution has (a, b, c) = (-1, -3, 5), with constant = 15 

So. with the broader definition, the terms now can become negative and the constant as well. 

 

If  a constant being small means absolute value small, then a good candidate is 

 constant =  -3

  17    -39     19

   1     -1     -3

 -21     37    -19

 or the negative of this. 

Here is a program and some results. The constant (row sum) is given first.

 

-60

  -1    -59      0

 -19    -20    -21

 -40     19    -39

 

-57

  -1    -57      1

 -17    -19    -21

 -39     19    -37

  

-45

   3    -53      5

 -13    -15    -17

 -35     23    -33

 

-21

  11    -45     13

  -5     -7     -9

 -27     31    -25

 

 -3

  17    -39     19

   1     -1     -3

 -21     37    -19

 

 -60

  -3    -57      0

 -17    -20    -23

 -40     17    -37

 

  -54

  -1    -55      2

 -15    -18    -21

 -38     19    -35

 etc...

 

        program ms

        implicit none

        integer a,b,c,j,con,ielt,flag,sign(3,9),term(9)

        data sign/0,-1,1,   1,1,1,   -1,0,1,

        1         -1,1,1,   0,0,1,   1,-1,1,

        2         1,0,1,   -1,-1,1,   0,1,1/

           do  a=-20,20

                do b=-20,20

                   do 1 c=-20,20

                        do ielt=1,9

        term(ielt)=sign(1,ielt)*a+sign(2,ielt)*b+sign(3,ielt)*c

                        call not_comp(term(ielt),flag)                        if(flag.eq.0)go to 1

                           do  j=1,ielt-1

                           if(term(ielt).eq.term(j))go to 1

                           enddo

                        enddo

                   con=3*c

                   print 3,con

3       format(/,i5)

                   print 2,term

2                  format( 3(i4,3x))

1                  enddo

                enddo

           enddo

        end

 

        subroutine not_comp(i,flag)

c Not composite means a positive integer not prime or 1

c Flag = 1 if true


        implicit none

        integer i,j,k,l,m,flag

        flag=1

        if (i.le.1)return

 

        k=sqrt(1.*i)

                do j=2,k

                m=(1.*i)/(1.*j)

                l=m*j

                if(l.eq.i)go to 1

                enddo

            flag=1

            return

1       flag=0

        return         

end

Edited on September 29, 2018, 7:41 pm
  Posted by Steven Lord on 2018-09-29 11:55:54

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