Pmath 340 Midterm Notes PDF

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PMATH 340 Midterm Notes Chapters: 2, 3, 10, 11, 16, 17, 20, 21, 22, 27, 28, 29 Assignments: Coverage: Pythagorean Triples, Euler Formula, Primitive roots and their applications, Quadratic Reciprocity 5 questions 2 will be taken directly from the homework Before Assignment 1 (Lectures Ch 2, 3, 10, 11...

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PMATH 340 Midterm Notes Chapters: 2, 3, 10, 11, 16, 17, 20, 21, 22, 27, 28, 29 Assignments: 1-5 Coverage: Pythagorean Triples, Euler Formula, Primitive roots and their applications, Quadratic Reciprocity 5 questions – 2 will be taken directly from the homework Before Assignment 1 (Lectures 1-5, Ch 2, 3, 10, 11, 16): Textbook Notes: a ≡b ( mod m ) ⇒m∨a−b=a-b divides m ⇒a=mx + b , for any x ∈ Z - Primitive Pythagorean Triple (PPT): A triple of numbers (a,b,c) such that a,b, and c have no common factors and satisfy a 2+b2 = c2 - One of a and b is odd, the other is even, and c is always odd - Theorem 2.1 (Pythagorean Triples Thoerem): We will get every primitive Pythagorean triple (a,b,c) with a dd and b even by using the formulas

a=st , b= -

s2 −t2 s 2 +t2 , c= 2 2

where s>t≥1 are chosen to be any odd integers with no common factors Theorem 3.1: Every point on the circle x 2 + y 2 =1 whose coordinates are rational numbers can be obtained from the formula

-

-

-

-

-

( x , y )=

(

1−m 2 2m , 2 1+m 2 1+ m

)

by substituting in rational numbers for m (except

for the point (-1,0) which is the limiting value as m→ ∞ ) Fermat’s Little Theorem: p −1 If p is a prime and p∤ a , then a ≡1(mod p) Euler’s Phi Function: ϕ( m )=¿ {a :1≤ a ≤ m∧gcd ( a , m) =1 } , ϕ( p )= p−1 for any prime p Theorem 10.1 (Euler’s Formula): If gcd(a,m)=1, then aϕ(m ) ≡ 1(mod m) Lemma 10.2: Let 1≤ b1...