Elliptic curves and 3x3 magic squares of squares
by Randall Rathbun, March 2017.

to Christian Boyer, March 11, 2017

I discovered this evening (2017-Mar-11 around 8 pm) that Andrew Bremner's example of 7 squares in his category VII for a 3x3 magic square of squares is actually only 1 point on a rank-4 elliptic curve:

  y^2 = x^3 + 680620676471634707316000000*x + 18254260159086640948557987978000000000000

This is rank 4 curve:

This was found by carefully transforming his equation (11) in his 2nd paper to an elliptic curve, by Cassel's transform.  Allan MacLeod wrote up a procedure on a mathwebsite on how this transform is done.

I have found 51 other magic squares in this family (the coefficients are large), but alas, only his example has a 7th square in it. Since this is a rank-4 curve, it has lots of rational points on it, so I am trying to create a family of about 200-300 points or so, and see if perhaps 1 other point might lead to the 2nd solution, with 7 squares square. I will keep you informed how this goes.  At least it satisfied my hunch that his square was on an infinite family of squares, which they are.

to Andrew Bremner, cc Christian Boyer, March 12, 2017
(Andrew Bremner = author of ref [1] "On squares of squares II", Acta Arithmetica, XCIX.3, (2001) pf 289)

I exhausted all point heights < 515+ on (2) looking for a possible candidate to your interesting (1) square[1], but nothing turned up at all.  This is a very interesting anomaly.

Equation (11) using the alpha/beta/gamma/delta solution of (1037,13,1,-391) gives the quartic Q

(1)    y^2 = 1508598062500*x^4 - 4532183544000*x^3 + 3017196125000*x^2 - 4532183544000*x + 1508598062500

and using a reduction of Ian McConnell, given in his book, or Allan MacLeod's reduction, both result in the same minimal elliptic curve

(2)    E(Q) = y^2 = x^3 - x^2 + 7528505392*x + 671534074163712

which is rank 4 (by mwrank) and torsion group Z2. The rank 4 meant that I found 1272 points of height <= 515+, but unfortunately only 1 gave 7 squares. I think the likelihood of possibly extracting one more point on (2) creating the 7th square is not worth the effort.

I am still running the general search of all rational solutions for (12) and plugging into (13a,b,c) but only square (1) keeps turning up for alpha/beta/gamma/delta <= 1260 which satisfy both.

to Christian Boyer, March 12, 2017

I have made a serious effort for solving your enigma one and uncovered a rank 4 elliptic curve, which both you and Dr Andrew Bremner now know, courtesy of my emails.

Alas, using 1272 rational points on the elliptic curve, with numbers into the thousands of digits, did NOT turn up another magic 3x3 square of 7 squares. I need to add, that I used Michael Stoll's ratpoints program to check ALL the other hyperelliptic curves in Bremner's 2nd paper, and the program failed to find a single rational point on any of those curves, despite even one interesting curve which showed promise. This is discouraging.

I consider Bremner's 2nd paper the last word on 3x3 magic squares of squares, as he pretty much exhausted the spigot of current math knowledge. It was some effort for me to take his solution and uncover the quartic and elliptic curve!

to Christian Boyer, March 13, 2017

The minimal elliptic curve connected with the 7 square solution, Category VII, equation (11) in Bremner's paper is

   (1)    E(Q): y^2 = x^3 - x^2 + 7528505392x + 671534074163712

The two points [-59528,-3525800] height 5.988906+ and [16889018/169,-108121359150/2197] height:6.161973+ on this curve create the 7-squares 3x3 magic square (2) below:

         |   373^2  289^2   565^2  |
   (2)   |  360721  425^2    23^2  |
         |   205^2  527^2  222121  |

37,572 points on this curve have been checked, up to height 421+, but only the two points given above produced a magic 3x3 square with 7 squares.

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