3x3 square constructed with the 6th triple of primitive pythagorean triangles having equal areas
by Lee Morgenstern, March 2017.

This is related to the 3x3 magic square of squares problem. It is a solution to one of the problems in D21 of Richard Guy's "Unsolved Problems in Number Theory". There used to be 5 known triples of primitive pythagorean triangles having equal areas. Now there is a 6th triple. Early March 2017, on sci.math, I found a note by Duncan Moore who writes:

"Using an algorithm which searches exhaustively over areas I've discovered the 6th triple of primitive Pythagorean triangles with equal area. The area is 9381843970167926138271390 and the generators are (352538,2999447), (1931103,2398838) and (3063347,3215070). The search took roughly 35GHz-days."

These are solutions to

• mn(m+n)(m-n) = rs(r+s)(r-s) = pq(p+q)(p-q)

where m,n and r,s and p,q are coprime odd-even pairs.

They have a direct bearing on the existence of a 3x3 magic square of distinct squares. A triple of PPTs having the same area can be made into a 3x3 semi-magic square of squares with one magic diagonal.

Based on a massive search with areas up to 6 x 10^23 with m,n values under 16 million, I have found that the best chance for a 9-square magic square of squares is one using only primitive pythagorean triangles.That's because whenever there were scaled triangles, the scaling nearly always violated a modular requirement. Apparently I should have searched just a bit higher, with areas up to 9.38 x 10^24, but I only had a 4 terabyte disk drive back then.

Here are some details of the new solution.

• 352538 = 2 x 359 x 491
  2999447 = 11 x 257 x 1061
  (2999447 + 352538) = 3351985 = 5 x 7 x 13 x 53 x 139
  (2999447 - 352538) = 2646909 = 3 x 3 x 19 x 23 x 673
• 1931103 = 3 x 3 x 19 x 23 x 491
  2398838 = 2 x 13 x 257 x 359
  (2398838 + 1931103) = 4329941 = 7 x 11 x 53 x 1061
  (2398838 - 1931103) = 467735 = 5 x 139 x 673
• 3063347 = 7 x 23 x 53 x 359
  3215070 = 2 x 3 x 3 x 5 x 139 x 257
  (3215070 + 3063347) = 6278417 = 19 x 491 x 673
  (3215070 - 3063347) = 151723 = 11 x 13 x 1061

Each group has the same factors making an area of

• 2x3x3x5x7x11x13x19x23x53x139x257x359x491x673x1061

It is interesting that, although the area is very large, none of the prime factors in the generators are very large. If a future search is done, perhaps it can be accelerated by restricting the size of the primes (but in a different way than your search).

So, when we put this together to make a 3x3 semi-magic square of squares, will it be fully magic?

• (19720769947309)^2   (6757561171393)^2  (11290071470263)^2
  (10987237357337)^2   (9483582546853)^2  (18745169816089)^2
   (7239541562993)^2  (20650330341071)^2   (9120965347253)^2

The magic sum is 562039114103450691451191099 for the rows, columns, and one diagonal.

The other diagonal has the sum 269815013768924501820610827.

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