As of 05Feb2007:

These are the Extra Problems for Prof. King's Number Theory course.

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E1:  A pythagorean-triple (a,b,c) comprises integers satisfying

         a^2 + b^2  =  c^2.

The "c" is the HYPOTENUSE of a right-triangle, and "a" and "b" are the
LEGS of the triangle.

Prove that oddly-many of the sides in (a,b,c) are divisible by five.

Prove that some leg must be a multiple of 3.

Prove that some leg must be a multiple of 4.
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E2: Using the Euclidean Algorithm (LBolt),  compute the GCD of

        [6x^3+5x^2+4]  and   [x^2-1]

in |Q[x].   These two polys are coprime, so LBolt will give you a
|Q[x]-unit on the penultimate line.

  Scale the unit and the mulipliers s_n and t_n, so as to produce
polys S and T with

        1  =  [6x^3+5x^2+4]*S(x)  + [x^2-1]*T(x) .
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E3: Consider a commutative ring G.  Prove that

	Units(G)  is disjoint from  Zero-divisors(G).
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E4: Consider a tuple of posints

        Mvec  :=  (M_1, M_2, ..., M_J)

and its product  P := M_1 * M_2 * ... * M_J.   Define the j-th
reduced-product

        R_j  :=  P/M_j .

Prove:  Rvec is a coprime-tuple IFF
           Mvec is a pairwise-coprime--tuple.

This latter means that for each pair i<j in [1 .. J], necessarily

        M_i cop M_j .
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E5: Suppose G and R  are rings.  Prove that

	Units(G × R)   =   Units(G) × Units(R).
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E6:
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E7:   You have posints Mi, integers Ti, and congruences OTForm

Ci:	x  =Mi=  Ti ,

for i=1,2,3,...,N.  You simply want to know if the N many congruences *have*
a comman soln x.
  As N->infty, please analyse the running-time of each of the following
algorithms:

Alg-A:  You fuse (C1)  with (C2), then fuse the result with (C3), ...,
stopping if fusion is impossible.

Alg-B:  For  each index-pair  i<j in [1 .. N], you test to see if

	Gcd(Mi, Mj)  divides  [Ti - Tj].
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E8:
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E9:
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E10:
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