Number Theory Jason Filippou CMSC250 @ UMCP 06-08-2016 Jason - PowerPoint PPT Presentation

Number Theory Jason Filippou CMSC250 @ UMCP 06-08-2016 Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 1 / 1

Outline Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 2 / 1

Definition (?) of Number Theory The Queen of Mathematics. Study of the integers and their generalizations (primes, rationals, etc) Used to be known as arithmetic , but nowadays arithmetic refers to first grade calculations. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 3 / 1

Short Historical Overview Short Historical Overview Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 4 / 1

Short Historical Overview Babylonians Figure 1: The Plimpton 322 Babylonian tablet This tablet (created circa 1800 BC ) contains a series of large Pythagorean triples ! Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 5 / 1

Short Historical Overview Babylonians Figure 1: The Plimpton 322 Babylonian tablet This tablet (created circa 1800 BC ) contains a series of large Pythagorean triples ! Triplets of integers a, b, c such that a 2 + b 2 = c 2 Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 5 / 1

Short Historical Overview Babylonians These guys just brute-forced those numbers Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 6 / 1

Short Historical Overview Babylonians (3, 4, 5), (5, 12, 13) (7, 24, 25), (8, 15, 17) . . . , . . . (36, 323, 325), (37, 684, 685) . . . , ... Currently believed that this plaque establishes the mathematical identity: � 1 x − 1 � 1 x + 1 �� 2 �� 2 � � + 1 = 2 x 2 x Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 7 / 1

Short Historical Overview Egyptians, Greeks We don’t know anything else about Babylonian Number Theory! Babylonian algebra and astronomy , on the other hand... Also, Egyptian astronomy, geometry. Greek geometry . Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 8 / 1

Short Historical Overview Greek philosophers The Greek mathematicians Pythagoras and Thales were influenced either by the Babylonians or the Egyptians, or both. Pythagorean theorem. Thales’ theorem. Figure 2: Pythagoras of Samos. Figure 3: Thales of Miletus. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 9 / 1

Short Historical Overview Euclid Euclid’s Elements contain the first set of axioms of Number Theory as we know it today. In chapters 21-34 of his 9th book of Elements , Euclid makes statements such as: “Odd times even is even” “If an odd number divides an even number, it also divides half of it.” √ The 10th book in Elements contains a formal proof that 2 is an irrational number. This discovery was very upsetting for the Greeks. Figure 4: Euclid of Alexandria Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 10 / 1

Short Historical Overview Chinese The Chinese Remainder Theorem, or The Mathematical Classic of Sun Tzu TM (not the famous military tactician), states: Suppose n 1 , . . . , n k are integers , pairwise co-prime. Then, for any given sequence of integers a 1 , . . . , a k , there exists an integer x which solves the following system of equations: x ≡ a 1 ( mod n 1 ) x ≡ a 2 ( mod n 2 ) . . . x ≡ a k ( mod n k ) Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 11 / 1

Short Historical Overview ? Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 12 / 1

Short Historical Overview Fermat’s Last Theorem Arguably, the most famous problem in the history of Mathematics. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 13 / 1

Short Historical Overview Fermat’s Last Theorem Arguably, the most famous problem in the history of Mathematics. Statement: There do not exist positive integers a, b, c that satisfy the equation: a n + b n = c n for values of n ≥ 3 . Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 13 / 1

Short Historical Overview Fermat’s Last Theorem Arguably, the most famous problem in the history of Mathematics. Statement: There do not exist positive integers a, b, c that satisfy the equation: a n + b n = c n for values of n ≥ 3 . Fermat claimed an “elegant solution”, for which “the margin of the text was too small”. Finally proven by Sir Andrew Wiles, September 1994, 357 years after its inception! Figure 6: Sir Andrew Wiles. Figure 5: Pierre de Fermat. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 13 / 1

Short Historical Overview A hard branch of Mathematics Take-home message: Number theory is hard ! Hard to learn the math to understand it, hard to properly follow the enormous string of proofs (see: Wiles’ 1993 attempt). In this module, we’ll attempt to give you the weaponry to master the latter! Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 14 / 1

Short Historical Overview Famous open problems Hodge Conjecture. Riemann Hypothesis. Birch & Swinnerton-Dyer Conjecture. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 15 / 1

Short Historical Overview Famous open problems Hodge Conjecture. Riemann Hypothesis. Birch & Swinnerton-Dyer Conjecture. Goldbach’s conjecture. Statement: Every even integer greater than 2 can be expressed as the sum of two primes. Currently holds up to 4 × 10 8 , but not proven formally. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 15 / 1

Basic Definitions Basic Definitions Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 16 / 1

Basic Definitions Commonly used number sets Naturals N Naturals without zero N ∗ N odd , N even , respectively Odd and even naturals Integers Z Integers without zero Z ∗ Z + Positive integers with zero (equiv. to naturals) Positive integers without zero (equiv. Z ∗ + to N ∗ ) Negative integers with or without zero Z − , Z ∗ − Z odd , Z even Odd and even integers Rational numbers Q Real numbers R R + , R − , R ∗ Positive, negative real numbers, with or + , R ∗ − without zero Prime numbers P Table 1: Some commonly used number set symbols. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 17 / 1

Basic Definitions Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2 k . a Common abbreviation for “if and only if”. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 18 / 1

Basic Definitions Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2 k . a Common abbreviation for “if and only if”. Corollary (Parity of 0) 0 is an even number. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 18 / 1

Basic Definitions Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2 k . a Common abbreviation for “if and only if”. Corollary (Parity of 0) 0 is an even number. Definition (Odd numbers) An integer n is odd iff there exists an integer k such that n = 2 k + 1. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 18 / 1

Basic Definitions Rational numbers Definition (Rational number) A number r is called rational iff ∃ m ∈ Z , n ∈ Z ∗ such that r = m n . Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 19 / 1

Basic Definitions Rational numbers Definition (Rational number) A number r is called rational iff ∃ m ∈ Z , n ∈ Z ∗ such that r = m n . Corollary (Integer are rationals) Every integer number is also rational. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 19 / 1

Basic Definitions Hierarchy of number sets ℚ ℤ ℕ Figure 7: Our current hierarchy of number sets. Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 20 / 1

Basic Definitions Rational questions on rational numbers for rational students Are the following numbers rational? Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 21 / 1

Basic Definitions Rational questions on rational numbers for rational students Are the following numbers rational? 2 / 3 1 Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 21 / 1

Basic Definitions Rational questions on rational numbers for rational students Are the following numbers rational? 2 / 3 1 20 / 30 2 Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 21 / 1

Basic Definitions Rational questions on rational numbers for rational students Are the following numbers rational? 2 / 3 1 20 / 30 2 2 ∗ 10 5 / 3 3 Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 21 / 1