Lecture 022

Number Theory


Prime: let n\in \mathbb{Z} s.t n\geq 2. n is prime iff its only positive divisors are 1 and itself.

Composite: n is not prime (ie. (\exists a,b \in \mathbb{Z})(1< a \leq b<n \land n=ab))

Theorem: if n is composite, then n has a prime factor p<=\sqrt{n}

Common Divisor: for a, b \in \mathbb{Z} not both 0, integer d is called a common divisor iff d|a and d|b. d is called the greatest common divisor(d=gcd(a, b)) iff d is the greatest of the common divisor

coprime(relatively prime): iff gcd(a,b) = 1

Theorem: for none zero a,b. d=gcd(a,b) then gcd(a/d, b/d) = 1

Euclidean Algorithm:

Division Algorithm:

Corollary to Division Algorithm (DA): let a, b, in \mathbb{Z} with b!=0. if a=bq+r, then gcd(a,b) = gcd(b,r)

Proposition: a|b \land a|c \implies a|bx+cy

Euclidean Algorithm (formal)

Bezout's Lemma

  1. (\exists x, y \in \mathbb{Z})(ax+by = gcd(a,b)))
  2. (x,y \in \mathbb{Z} \land ax+by>0 \implies ax+by \geq gcd(a,b)) TODO: proof ignored

  3. corollaries: let a,b \in \mathbb{Z}, not both 0. let d=gcd(a,b) then

    1. if t \in \mathbb{Z} s.t. t|a and t|b, then t|d. (greatest common divisor divides any common divisor)
    2. gcd(a,b) | c \iff (\forall c \in \mathbb{Z})(\exists m, n \in \mathbb{Z})(c=am + bn)
    3. if a and b are relatively prime, then (\exists m, n \in \mathbb{Z})(am+bn=1)
    4. (\forall m \in \mathbb{Z}^+)(gcd(ma, mb)= m \times gcd(a,b))

TODO: proof ignored

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