Three new searches for a magic hourglass
by Lee Morgenstern, June 2014.

I have just completed three new searches for a Magic Hourglass.

The first search was for

• [m²+n²)(r²+s²)]² +- 4[mn(m²-n²)(r²+s²)² + rs(r²-s²)(m²+n²)²] both squares or
• [m²+n²)(r²+s²)]² +- 4[mn(m²-n²)(r²+s²)² - rs(r²-s²)(m²+n²)²] both squares for all values of m,n,r,s < 9000, with m,n coprime and r,s coprime.

The second search used the same formulas as above, but with

• (m²+n²)(r²+s²) < 5 x 10^12, with m,n coprime and r,s coprime.

The third search was for

• (m²+n²)² +- 4[mn(m²-n²) + rs(r²-s²)] both squares or
• (m²+n²)² +- 4[mn(m²-n²) - rs(r²-s²)] both squares, for (m²+n²) = (r²+s²) < 5 x 10^12, and m,n and r,s could have factors in common.

No solutions were found.

The third search was similar to the Buell search and in the same range, except that instead of using triples of m²+n² = r²+s² = u²+v², which restricts the possible scalings, I used pairs, which allows m²+n² = r²+s² = (u²+v²)t, where t can be a non-square. So my search tested more possibilities than the Buell search and also serves as a verification that the Buell search correctly found no solutions.

The second search was the opposite extreme to the Buell search. Instead of m²+n² = r²+s², it was for m²+n² having no factors in common with r²+s². This requires (u²+v²)t to be the product of the other two sums of squares.

The first search was for all central values less than about 8.1 x 10^7 and a partial search for central values up to almost (2x9000²)², which is about 2.6 x 10^16, thus there was the possibility of finding a solution where the square of the central value could be up to 33 digits long.

The second and third search both took about 10 hours running on four i7 processors. The first search took several weeks.