# report on sequences from magic squares of primes

Richard Guy rkg at cpsc.ucalgary.ca
Thu Aug 29 18:32:02 CEST 2002

```Dudeney, on p. 125 of Amusements in Mathematics,
says that the problem of constructing magic squares
with prime numbers was first discussed by myself in
`The Weekly Dispatch'for 22nd July and 5th August
1900; but during the last three or four years
[his preface is dated March 25, 1917]
it has received great attention from American
mathematicians.

I won't qoute the lot, but he concludes with a table
taken from `The Monist' (Chicago) for October 1913.

For further details the reader should consult
the article itself, by W. S. Andrews & H. A. Sayles.

On Thu, 29 Aug 2002, N. J. A. Sloane wrote:

> Thanks to Divid Wilson for pointing out Harvey Heinz's very nice web page.
>
> The OEIS now contains the following:
>
> %I A073520
> %S A073520 2,0,4440084513,258,1703,930
> %N A073520 Smallest magic constant for any n X n magic square made from consecutive primes, or 0 if no such magic square exists.
> %D A073520 H. L. Nelson, Journal of Recreational Mathematics, 1988, vol. 20:3, p. 214.
> %H A073520 Harvey Heinz, <a href="http://www.geocities.com/CapeCanaveral/Launchpad/4057/primesqr.htm">Prime Magic Squares</a>
> %H A073520 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#magic">Index entries for sequences related to magic squares</a>
> %O A073520 1,1
> %Y A073520 For the magic squares of orders 3 through 6 see A073519, A073521, A073522, A073523.
> %K A073520 nonn,nice,more,hard
> %A A073520 njas, Aug 29 2002
>
> If the primes don't need to be consecutive then all I know is that
> the sequence begins 2,0,177.  Can anyone supply more terms?
>
> If 1 is counted as a prime, and the primes don't need to be consecutive, then we have:
>
> %I A073502
> %S A073502 111,102,213,408,699,1114,1681,2416
> %N A073502 Magic constant for n X n magic square with prime entries (regarding 1 as a prime) with smallest row sums.
> %C A073502 Until the early part of the twentieth century 1 was regarded as a prime (cf. A008578).
> %C A073502 I don't know how far these entries have been rigorously computed. Lee Sallows has confirmed the first term (cf. A073473).
> %D A073502 W. S. Andrews and H. A. Sayles, The Monist (Chicago) for October 1913.
> %D A073502 H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 125, who quotes the Andrews-Sayles article as his source.
> %H A073502 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#magic">Index entries for sequences related to magic squares</a>
> %Y A073502 Cf. A073473 (for the n=3 square), A024351.
> %K A073502 nonn,new
> %O A073502 3,1
> %A A073502 njas, Aug 27 2002
> %E A073502 What is the analogous sequence when 1 is not allowed? All I know is that it begins with 177 (see A024351).
> %E A073502 Dudeney gives 36095/11 for n = 11 (an obvious typo) and 4514 for n = 12.
>
> Only the first term has been confirmed - can anyone help
> with subsequent terms?
>
> Neil Sloane, njas at research.att.com
>
>
>
>

```