smallest all-semiprime magic square

[Bob Barbour]

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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