Number Theory. Jason Filippou UMCP. ason Filippou UMCP)Number Theory History & Definitions / 1

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1 Number Theory Jason Filippou UMCP ason Filippou UMCP)Number Theory History & Definitions / 1

2 Outline ason Filippou UMCP)Number Theory History & Definitions / 1

3 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. ason Filippou UMCP)Number Theory History & Definitions / 1

4 Short Historical Overview Short Historical Overview ason Filippou UMCP)Number Theory History & Definitions / 1

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

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

7 Short Historical Overview Babylonians These guys just brute-forced those numbers ason Filippou UMCP)Number Theory History & Definitions / 1

8 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 2( 1 )) 2 ( = x + x 2( 1 )) 2 x ason Filippou UMCP)Number Theory History & Definitions / 1

9 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. ason Filippou UMCP)Number Theory History & Definitions / 1

10 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. ason Filippou UMCP)Number Theory History & Definitions / 1

11 Short Historical Overview Euclid Euclid s Elements contain the first set of axioms of Number Theory as we know it today. In chapters 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 ason Filippou UMCP)Number Theory History & Definitions / 1

12 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 ) ason Filippou UMCP)Number Theory History & Definitions / 1

13 Short Historical Overview? ason Filippou UMCP)Number Theory History & Definitions / 1

14 Short Historical Overview Fermat s Last Theorem Arguably, the most famous problem in the history of Mathematics. Jason Filippou UMCP)Number Theory History & Definitions / 1

15 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: an + bn = cn for values of n 3. Jason Filippou UMCP)Number Theory History & Definitions / 1

16 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: an + bn = cn 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 5: Pierre de Fermat. Figure 6: Sir Andrew Wiles. Jason Filippou UMCP)Number Theory History & Definitions / 1

17 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! ason Filippou UMCP)Number Theory History & Definitions / 1

18 Short Historical Overview Famous open problems Hodge Conjecture. Riemann Hypothesis. Birch & Swinnerton-Dyer Conjecture. ason Filippou UMCP)Number Theory History & Definitions / 1

19 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 , but not proven formally. ason Filippou UMCP)Number Theory History & Definitions / 1

20 Basic Definitions ason Filippou UMCP)Number Theory History & Definitions / 1

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

22 Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2k. a Common abbreviation for if and only if. ason Filippou UMCP)Number Theory History & Definitions / 1

23 Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2k. a Common abbreviation for if and only if. Corollary (Parity of 0) 0 is an even number. ason Filippou UMCP)Number Theory History & Definitions / 1

24 Parity Definition (Even numbers) An integer n is even iff a there exists an integer k such that n = 2k. 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 = 2k + 1. ason Filippou UMCP)Number Theory History & Definitions / 1

25 Rational numbers Definition (Rational number) A number r is called rational iff m Z, n Z such that r = m n. ason Filippou UMCP)Number Theory History & Definitions / 1

26 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. ason Filippou UMCP)Number Theory History & Definitions / 1

27 Hierarchy of number sets Q Z N Figure 7: Our current hierarchy of number sets. ason Filippou UMCP)Number Theory History & Definitions / 1

28 Rational questions on rational numbers for rational students Are the following numbers rational? ason Filippou UMCP)Number Theory History & Definitions / 1

29 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 ason Filippou UMCP)Number Theory History & Definitions / 1

30 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/30 ason Filippou UMCP)Number Theory History & Definitions / 1

31 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 ason Filippou UMCP)Number Theory History & Definitions / 1

32 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/1 ason Filippou UMCP)Number Theory History & Definitions / 1

33 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/ ason Filippou UMCP)Number Theory History & Definitions / 1

34 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/ ason Filippou UMCP)Number Theory History & Definitions / 1

35 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/ ason Filippou UMCP)Number Theory History & Definitions / 1

36 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/ π, φ, e, 2 ason Filippou UMCP)Number Theory History & Definitions / 1

37 Rational questions on rational numbers for rational students Are the following numbers rational? 1 2/3 2 20/ /3 4 0/ π, φ, e, 2 So there s something beyond rational numbers! ason Filippou UMCP)Number Theory History & Definitions / 1

38 Irrational numbers Definition (Irrational numbers) A number is irrational iff it is not rational. ason Filippou UMCP)Number Theory History & Definitions / 1

39 Irrational numbers Definition (Irrational numbers) A number is irrational iff it is not rational. Corollary If s is an irrational number, there does not exist a pair of integers m, n, with n 0, such that s = m n. ason Filippou UMCP)Number Theory History & Definitions / 1

40 Irrational numbers Definition (Irrational numbers) A number is irrational iff it is not rational. Corollary If s is an irrational number, there does not exist a pair of integers m, n, with n 0, such that s = m n. Rationals and irrationals together give us the set of real numbers: R. ason Filippou UMCP)Number Theory History & Definitions / 1

41 Hierarchy of number sets, revisited R Q Z R Q N Figure 8: Our hierarchy, updated. We will not deal with higher number systems (complex numbers, quaternions). Note that rationals and irrationals complement each other (obviously). ason Filippou UMCP)Number Theory History & Definitions / 1

42 Prime numbers Definition (Prime number) An integer n 2 is called prime iff its only factors (divisors) are 1 and n. ason Filippou UMCP)Number Theory History & Definitions / 1

43 Prime numbers Definition (Prime number) An integer n 2 is called prime iff its only factors (divisors) are 1 and n. Corollary (Primality of 2) 2 is the only even prime number. ason Filippou UMCP)Number Theory History & Definitions / 1

44 Prime numbers Definition (Prime number) An integer n 2 is called prime iff its only factors (divisors) are 1 and n. Corollary (Primality of 2) 2 is the only even prime number. Prime numbers are fundamental in Number Theory, for a variety of reasons. Important enough that the set of primes has a symbol: P Largest known prime: 2 74,207,281 1 (22,338,618 digits). Discovered by the Great Internet Mersenne Prime Search (GIMPS). ason Filippou UMCP)Number Theory History & Definitions / 1

45 Composite numbers Definition (Composite number) An integer n 2 is called composite iff it is not prime. ason Filippou UMCP)Number Theory History & Definitions / 1

46 Composite numbers Definition (Composite number) An integer n 2 is called composite iff it is not prime. Corollary (Primality of 0 and 1) 1 and 0 are neither prime nor composite. ason Filippou UMCP)Number Theory History & Definitions / 1

47 Hierarchy of number sets, revisited R Q Z R Q N P Figure 9: Our final hierarchy. Possible to define more sets, like even and odd integers, Mersenne primes, etc. ason Filippou UMCP)Number Theory History & Definitions / 1