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       #Post#: 116--------------------------------------------------
       SUMS OF 3 AND 4 SQUARES
       By: eba95 Date: July 30, 2010, 6:48 am
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       Sums of Three and Four
       Squares
       How many squares does it take to
       express every whole number as the
       sum of squares? We saw that two
       was not enough in Sums of Two
       Squares. Perhaps three? Or four?
       Well, three is not enough, but
       almost. The only whole numbers
       which cannot be written as the sum
       of 3 squares are numbers of the form
       4m(8k+7). So you will have problems
       writing 7, 15, or 28 as the sum of
       three squares.
       But every whole number can be
       written as the sum of four squares!
       Accordingly, 7=22+12+12+12, and
       15=32+22+12+12.
       Presentation Suggestions:
       Have the class pick their favorite
       number and write it as the sum of
       four squares.
       The Math Behind the Fact:
       The sum of 4 squares result was
       stated by Gerard, Fermat, and
       Diophantus(?), but first proved by
       Lagrange in 1770.
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