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Q3: Given an integer tuple xvec=(a, b, c, d) of any length, let
sumsqs(xvec)
mean the "sum of squares" a²+b²+c²+d².
The LAGRANGE 4SQUARE THEOREM says that for each posint K, there
exist a natnum 4-tuple xvec [some entries might be zero] so that
sumsqs(xvec) = K.
Part of the proof involves "melding" the polynomial a²+b²+c²+d²,
analogously to our melding of poly x²+y².
DEFINITION: Given 4-tuples
xvec := (x1, x2, x3, x4) and yvec := (y1, y2, y3, y4),
define the "Lagrange meld"
1: zvec := Lagmeld(xvec, yvec)
as shown on page NZM317, below the statement of Thm6.26. Here,
z1 is the contents of the first pair of parentheses, on the
RHSide of the equality. Etc.
Q3a: Note that
155 = 9²+7²+5² and
57 = 2²+4²+6²+1² .
Please write 8835 [= 155*57] as a sum of four squares,
showing the interesting aspects of the lagmelding.
Q3b: Do something interesting with lagmelding; make up an
interesting problem and then solve it.
Q3c: Give a formal proof, from (1), that
sumsqs(zvec) = sumsqs(xvec)·sumsqs(yvec).
[NZM does not give a proof.] You may want to break your
argument into a lemma or two.
Aside for-the-Curious: Note that x²+y² is the "norm" of the
complex number x+yi. [By "norm" I mean here the square of
absolute value.] Thus that "melding is sealed under
multiplication" is simply the statement that the abs-value of a
product [of complex numbers] is the product of their
abs-values.
It turns out that
a²+b²+c²+d²
is the norm of the "quaternion" a+bi+cj+dk. [Of course, you don't
need to know this in order to do the problem.]
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