Mathematics 466 Homework (due Sep. 3) A. Hulpke
1)
I will assume that you know the following topics from the introduction. (This is material you will have seen when you took M360, M366 or M369.) If not, please read the appropriate sections, and ask (in class or during office hours) if anything is not clear. (Nothing needs to be handed in for this problem.)
Notations 1.1
Integers, Rationals, Reals, Complex Numbers 1.4
Polynomial Division
Proofs
1.11
1.2
Induction
Sets
Basic Functions
Matrices and Determinants
1.5
1.13,1.14
1.15
1.23,1.24
2) a) Solve the equation
(
7 x
+
15
) mod 31
=
14.
b) Find (for example by trying out all values of x ) all solutions to the equation x c) Find all solutions to the equation x
2 mod 7
=
3.
d) Find all solutions to the equation x
2 mod 8
=
2 mod 7
=
1. Why are there more than two solutions?
2.
3) Show that it is not possible to define a division with remainder (I.e. write a
= qb
+ r with r
< b ) in the set of polynomials in two variables. (Hint: What would be the remainders when dividing x by y , and when dividing y by x ?
What does this imply for the ordering?)
(
Note:
This means we cannot use the Euclidean algorithm for multivariate polynomials. Nevertheless it makes sense to talk about a gcd of multivariate polynomials, we just can’t find it easily.)
4)
Calculate the gcd of x
4
+
4 x
3
−
13 x
2
−
28 x
+
60 and x
6
+
5 x
5
− x
4
−
29 x
3
−
12 x
2
+
36 x
.
5)
Show that there must be a power of 3 (with exponent
Hint: consider the sequence 3 mod 10000, 3 2
≥ mod 10000, 3
1, i.e. we exclude 3
3 mod 10000, 3 4
0 ) whose last four digits are 0001.
mod 10000, 3 5 mod 10000, . . . Show that this sequence must be eventually periodic. Then using the invertability of 3 modulo 10000 show that one can go backwards.
6 ∗ )
The easter formula of Gauß provides a way for calculating the date of easter (i.e. the first sunday after the first full moon of spring) in any particular year. For the period 1900-2099 (a restriction due to the use of the Georgian calendar) a simplified version for year
Y is:
• Calculate a
=
Y mod 19, b
=
Y mod 4, c
=
Y mod 7.
• Calculate d
= (
19 a
+
24
) mod
30 and e
= (
2 b
+
4 c
+
6 d
+
5
) mod
7
• If d
+ e
<
10 then easter will fall on the
( d
+ e
+
22
) th March, otherwise the
( d
+ e
−
9
) th of April.
• Exception: If the date calculated is April 26, then easter will be April 19 instead. (There is a further exception in western (i.e. not orthodox) churches for certain April 25 dates, again due to the Georgian calendar, which we ignore for simplicity.) a) Calculate the easter day for 2008 and for 1968.
b) Show (possibly by explicity calculation) that in your life time 1 Easter will never be earlier than this year.
c) Assuming the formula would work for any year (not just 1900-2099), what is the smallest period, after which we can guarantee that the sequence of easter dates repeats?
Problems marked with a
∗ are bonus problems for extra credit.
1 Assuming there are is no unforseen progress in medicine, i.e. until 2099