smallest all-semiprime magic square

Sun Dec 24 11:17:43 CET 2006

Thank you, Bob. Me, my, son, and my wife (who has a sister farming in New
Zealand) all think that Peter Loly is doing wonderful work.  Just a few
hours ago, I submitted two sequences citing him, one of them being as below.

%I A126276
%S A126276 18, 504, 5200, 31500, 136710, 471968, 1378944
%N A126276 Moment of inertia of all magic cubes of order n.
%C A126276 Loly and Rogers show that the moment of inertia of a magic
cube is n^3(n^3 + 1)(n^2 - 1)/12. In effect, they demonstrate that
magic cubes have the same inertial form as a spherical top.
Loly investigated the "physical" properties of magic squares--treating
the numbers of each such square as physical quantities. If the integers
are consecutive numbers from 1 to n^2, the square is said to be of nth
order. The magic sum itself is given by n(n^2 + 1)/2. Suppose you
interpret the numbers as masses. You can then determine a magic square's
moment of inertia about a given axis of rotation. For any specific case,
you obtain the moment of inertia, In, of a magic square of order n about
an axis at right angles to its center by summing mr^2 for each cell,
where m is the number centered in a cell and r is the distance of the
center of that cell from the center of the square measured in units of the
nearest neighbor distance.
You find that the moment of inertia, I_z, about the square's center (an
axis at right angles to the square) is twice the moment of inertia
about an axis of rotation along the center row or column.
Such an analysis can be extended to magic cubes. A magic cube consists
of n^3 numbers, arranged so that each row, column, and main diagonal
give the same sum. In the case, the magic constant is n(n^3 + 1)/2.
%H A126276 Ivars Peterson, <a
href="http://www.sciencenews.org/articles/20060701/mathtrek.asp">Magic
Square Physics</a>. Science News online, Week of July 1, 2006; Vol.
170, No. 1
%H A126276 Peter Loly, <a
href="http://home.cc.umanitoba.ca/~loly/MathGaz.pdf
<http://home.cc.umanitoba.ca/%7Eloly/MathGaz.pdf>">The invariance
of the
moment of inertia of magic squares</a>, Mathematical Gazette
88(March 2004):151-153
%H A126276 Adam Rogers and Peter Loly, <a
href="http://www.cupj.ca/0302_rotational.pdf">Rotational sorcery: The
inertial properties of magic squares and cubes</a>. Canadian
Undergraduate Physics Journal 3(No. 2):25, 2005.
%F A126276 a(n) =  (n^3)*(n^3 + 1)*(n^2 - 1)/12.
%Y A126276 Cf. A126276 Moment of inertia of all magic cubes of order n
%O A126276 2,1
%K A126276 ,easy,nonn,
%A A126276 Jonathan Vos Post (jvospost2 at yahoo.com
<http://us.f551.mail.yahoo.com/ym/Compose?To=jvospost2@yahoo.com&YY=11658&y5beta=yes&y5beta=yes&order=down&sort=date&pos=1&view=a&head=b>),
Dec 23 2006
RH
RA 192.20.225.32
RU
RI

On 12/24/06, Bob Barbour <bbarbour at unitec.ac.nz> wrote:
>
> Hi Jonathan and seqfans,
>
> Seasons Greeting from New Zealand.
> Peter Loly and his team have done substantial work
> enumerating magic squares.
>
> loly at cc dot umanitoba dot ca
>
> should get to him.
>
> Regards,
> Bob
>
> >>> "Jonathan Post" <jvospost3 at gmail.com> 12/24/06 9:46 PM >>>
> Dear Frank,
>
> It does look as if I imposed an extra constraint on constructing
> semiprime
> magic squares. Interesting that it did not overconstrain and prevent
> solution.
>
> So now we have five open problems:
>
> (1) How many different semiprime nxn magic squares are there with
> maximum
> element < k;
>
> (2) What is the sequence of row sums (magic numbers) of my semiprime
> magic
> squares whose elements are in arithmetic sequence;
>
> (3) Tony Noe's question: what is the smallest nxn semiprime magic square
> where all elements are relatively prime;
>
> (4) My modification of the previous: what is the smallest nxn semiprime
> magic square where all elements in the same row, column, or diagonal are
> relatively prime;
>
> (5) Dr. Geoffrey Landis' question: what is the smallest semiprime magic
> square whose magic number is semiprime?
>
> Making a simple problem hard seems to be one of my specialties;)
>
> Happy holidays,
>
> Jonathan Vos Post
>
>
>
> On 12/23/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
> >
> > The smallest prime magic square is:
> >
> > 17 89  71
> > 113 59   5
> > 47 29 101
> >
> > We can just double that, to get:
> >
> > 34 178 142
> > 226 118  10
> > 94  58 202
> >
> > which is quite a bit smaller than your example.
> >
> > Even if you want to exclude squares where the entries have a common
> > divisor, your example looks suspiciously large to me.  And looking at
> > A096003, it looks like you are assuming that your magic consists of
> > numbers in arithmetic progression.  That is not a valid assumption.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: jvospost3 at gmail.com
> >
> >    Using A096003 and A097824, here is the smallest all-semiprime magic
> > square, which I just discovered today:
> >
> > ============
> > 1139 635 995
> > 779 923 1067
> > 851 1211 707
> > ============
> >
> > In numerical order, these entries are:
> > 635 = 5*127
> > 707 = 7*101
> > 779 = 19*41
> > 851 = 23*37
> > 923 = 13*71
> > 995 = 5*199
> > 1067 = 11*97
> > 1139 = 17*67
> > 1211 = 7*173
> >
> >   As I just noted in a submission to Prime Curios: 2769 is the sum of
> > any row, column, or diagonal. Curiously, 2769 = 3 * 13 * 71, all three
> > of which are primes when reversed (but that is too "base" for us
> here).
> >
> >   Using A096003(16) = 28213, and A097824(16) = 354, one likewise has
> the
> > smallest semiprime magic square of order 4.
> >
> >   Using A096003(25) = 2012771, and A097824(25) = 9600, one likewise
> has
> > the smallest semiprime magic square of order 5.
> >
> >   I'm not sure if it's better to submit the finite full sequence 1139,
> > 635, 995, 779, 923, 1067,
> >   851, 1211, 707 for the order 3, and similarly for the order 4 and
> > order 5; or give the sequence of row sums as a function of order n; or
> > what.
> >
> > I don't know the order 6 or above.
> >
> >   I also know the smallest order 3 of the 3-almost prime magic
> squares,
> > and so forth, but these become "less" for most seqfans, I suspect.
> >
> >   I admit to not yet having written down the smallest semiprime magic
> > cbe, but the same method can be adapted...
> >
> > This is, new, is it not?
> >
> > Happy Hanukkah, Merry Christmas, etcetera,
> >
> > -- Jonathan Vos Post
> >
> >
> >
> >
> >
> ________________________________________________________________________
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> >
> >
>
>
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