smallest all-semiprime magic square

[Jonathan Post]

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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