As of 05Feb2007:
These are the Extra Problems for Prof. King's Number Theory course.
/--------------------------------------------------------\
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.
\________________________________________________________/
/--------------------------------------------------------\
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) .
\________________________________________________________/
/--------------------------------------------------------\
E3: Consider a commutative ring G. Prove that
Units(G) is disjoint from Zero-divisors(G).
\________________________________________________________/
/--------------------------------------------------------\
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 iinfty, 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