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<p class=MsoNormal><span style='color:#1F497D'>> In fact, it looks as if S =
{0} U {(4^j)(8k+1): j >= 0} . Presumably this would extend to the
nonnegative integers.<o:p></o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'>It should read “negative
integers”.<o:p></o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'>OK, I found something really
interesting.<o:p></o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='color:#1F497D'>Using the characterization S =
{0} U {(4^j)(8k+1): j >= 0}, the negative values of S are the negations
of A004215, the values that require 4 squares. If this characterization is
true, then n > 0 requires 4 squares precisely when –n is a square mod 2^k
for every k >= 0. I struggle to grok this.<o:p></o:p></span></p>
<p class=MsoNormal><o:p> </o:p></p>
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