Number theory, geometry and algebra - Dynamics-approx.jku.at

1 Number theory
1.1 Prime numbers and divisibilty:
We shall work with the following number systems:
N = {1,2,3,...} (1)
N 0 = {0,1,2,3,...} (2)
Z = N 0 ∪{−1,−2,−3,...}. (3)
We regard them as algebraic systems with the operations + und . of
addition and multiplication.
(N,+) is a commutative semigroup (without unit);
(N,.) is a commutative semigroup with unit 1;
(N 0 ,+) is a commutative semigroup with unit 0;
(Z,+,.) is a ring.
(See the last chapter for these notions).
Divisibility: If we have two numbers a,b ∈ Z, then we say that a is a
divisor of b (written: a|b) if there exists r ∈ Z with a = rb. If further
a ≠ b, then a is a proper divisor of b. The following simple properties are
self-evident:
a) If a|b i (i = 1,...,n), c 1 ,...,c n ∈ Z, then a| ∑ c i b i ;
b) b|a and c ∈ Z ⇒ bc|ac;
c) ac|bc (a,b,c ∈ Z, c ≠ 0) ⇒ a|b;
d) b|a ⇒ |b| ≤ |a| (a,b ∈ Z, a ≠ 0);
e) a|b and b|a ⇒ |a| = |b| (a,b ∈ Z\{0});
f) a|b, b|c ⇒ a|c (a,b,c ∈ Z).
The division algorithm: Suppose a ∈ Z, b ∈ N. Then there are unique
numbers q ∈ Z and r ∈ N 0 with r < b, so that a = qb+r
Proof. Existence: q is the largest number in the set {t ∈ Z : t ≤ a }. For
b
q ≤ a < q+1 and so qb ≤ a < b(q+1). Then let r = a−bq so that 0 ≤ r < b
b
and a = qb+r.
Uniqueness: Let a = qb+r = q 1 b+r 1 with r 1 ≥ r. Then (q−q 1 )b = r 1 −r.
But 0 ≤ r 1 −r < b. Thus q = q 1 etc.
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