smallest all-semiprime magic square

[Jonathan Post]

Here's a semiprime magic cube, but it is not the smallest. The smallest
order 3 prime cube is known, but not to me. The below is simply double the
elements of the Akio Suzuki 1977 prime cube shown in
http://members.shaw.ca/hdhcubes/cube_prime.htm
Thus it is very far indeed from coprimality within the rows, columns, or
squares.

4306...1858.....454,,,,,,,,,1018...3214...2386,,,,,,,,,,1294...1436...3778
1678...1894...3046,,,,,,,,,3574...2206.....838,,,,,,,,,,1366...2518...2734
..634...2866...3118,,,,,,,,,2026...1198...3394,,,,,,,,,,3958...2554.....106

The prime cube of which this is the double is described:

"Also constructed by Akio Suzuki  in 1977 [1].  This cube has exactly the
same characteristics as the above cube except it uses a smaller prime
numbers.  Prime numbers used range from 53 to 2153. "
"Each complement pair sums 2206 which is the sum of the smallest and largest
number used. In both cases, the middle number of the cube is this sum
divided by 2. Both of these characteristics are common to all associated
magic hypercubes. The constant is 3309."

   *Addendum:* As a result of a computer search, Allen Wm, Johnson, Jr. [2]
confirmed that this cube has the smallest possible sum for an order 3 prime
magic cube using distinct digits.

[1] Gakuho Abe, *Related Magic Squares with Prime Elements*, JRM 10:2
1977-78, pp.96-97. Akio Suzuki order-3 and 4 cubes.

[2] A. W. Johnson, Jr., *Solution to Problem 2617*, JRM 32:4, 2003-2004, pp.
338-339

[3] A. W. Johnson, Jr., *An Order 4 Prime Magic Cube,* JRM 18:1, 1985-86, pp
5-7


On 12/24/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> Has progress been made since this article?
>
> Magic Tesseracts
> Ivars Peterson
> http://www.sciencenews.org/pages/sn_arc99/10_16_99/mathland.htm
>
> The smallest perfect magic tesseract is or order 16 (i.e. 16 x 16 x 16 x
> 16).
>
> Are there prime magic tesseracts known? Can we construct semiprime magic
> tesseracts analogous to the semiprime magic squares discussed earlier?
>
> Does the Peter Loly result on moment of intertia of magic squares and
> magic cubes extend to magic tesseracts, with modifications since 4-D
> rotation is about a plane rather than about an axis?
>
> On 12/24/06, David Wilson <davidwwilson at comcast.net > wrote:
> >
> >  Oh wait, I just had a "duh" moment.
> >
> > For a 3x3 magic square with center entry k, the row sum is 3k. So for a
> > 3x3 prime magic square, the row sum is 3*prime, for a semiprime magic
> > square, the row sum is 3*semiprime, for a parition number magic square, the
> > row sum is 3*partition number, etc.
> >
> > Thus the 3x3 semiprime magic square cannot have a semiprime sum.
> >
>
>
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