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  1.  
    "Invert and multiply!" he cried, as the registers creaked and the half-adders rolled and swung. "Invert and multiply!" But it was too late, for the accumulators were dry and the buffers overflowing.
  2.  
    Multiply two numbers, each of which is the sum of two perfect squares. The result will also be the sum of two perfect squares.

    Funky, hunh?
    Brahmaputra - 600 ish AD
    Gauss - 20 page proof
    Bhargava - tiny proof very recently.

    So don't try this at home.
  3.  
    For example

    22 + 32 = 13
    42 + 52 = 41

    13 * 41 = 533
    = 232 + 22
    = 222 + 72

    Magic innit
  4.  
    What's described above is a special case of a far grander fundamental structure.

    The most trivial generalisation was discovered way back by Brahmaputra, who found that, for all integer variables:

    if for any k > 0
    n0 = a2 + k*b2
    n1 = c2 + k*d2
    then
    n0*n1 = x2 + k*y2

    The example given previously is a special case k=1 of this more general rule.

    Gauss generalised on Brahmaputra, and Bhargava generalised on Gauss.
    https://www.jstor.org/stable/3597249
    •  
      CommentAuthoroak
    • CommentTimeMay 22nd 2019