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Q0: Make up your own interesting, elegant, NT problem, then solve it.
Aesthetics counts.
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Show all reasoning for the following problem. Explain computations
if any part is obscure. Review the Chinese Remainder Theorem, CRT.
Q1: Define 4pos primes
P1 := 13, P2 := 17, P3 := 29,
and corresponding ronos
I1 := 5, I2 := 4, I3 := 12.
Let M be the product P1·P2·P3 =NOTE= 6409.
Q1a: Use CRT to construct four distinct integers
0 < A < B < C < C < M/2
so that each is an M-rono. [In particular, A² =M= -1.]
[Note: Just show the computations for one rono, A, then show me the
result, but not the computations, for B,C,D.]
Observe that each of ±Ij is a soln to polynomial congruence
x²+1 == 0 (mod Pj).
Thus CRT will give you 2·2·2=8 solutions [as M-symm-residues] to
x²+1 == 0 (mod M).
And four of these solutions will be positive.
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Q1b: Produce four REALLY different intpairs (x, y) such that
x² + y² = 6409 .
CHOICE: You may either apply SOTS-Euclidean-Algorithm to each of your
four M-rono. OR, you can meld together solutions to x² + y² = Pj, for
the various primes. Whatever method you choose, explain it carefully
for one computation.
Question: If you do both SOTS-Euclidean-Algorithm and melding, do you
get the same four solutions? ["Same", up to re-ordering and sign change,
naturally.]
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