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Number Theory

Number Theory Number Theory: A reflection of the basic mathematical endeavor. Exploration Of Patterns: Number theory abounds with patterns and requires little background to understand questions. Inductive Reasoning: Patterns are discovered and generalized.

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Number Theory

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  1. Number Theory Number Theory: A reflection of the basic mathematical endeavor. Exploration Of Patterns: Number theory abounds with patterns and requires little background to understand questions.

  2. Inductive Reasoning: Patterns are discovered and generalized. • Deductive Reasoning: Patterns are formalized and verified with proof.

  3. Pythagoras of Samos Born: about 569 BC in Samos, IoniaDied: about 475 BC

  4. Pythagorean Society • 4000 years ago: Traders, calender makers and surveyors used large natural numbers. • 500 B.C: Pythagorean school considered the natural numbers to be the key to understanding the universe. -Basis of philosophy and religion -Imparted to them humanistic and mystic properties.

  5. Natural numbers were their friends, associates, tools and enemies. • Applied adjectives associated with people such as friendly, perfect, natural, rational. • Disintegration of school almost occurred when they discovered that not all physical quantities were expressible as ratios of natural numbers.

  6. Discovery of 2 as irrational number 2 1 1

  7. Perfect Numbers • A natural number is perfect if it is the sum of its divisors. • Ancient Greeks knew 4 perfect numbers and endowed them with mystic properties.

  8. 6 = 3 + 2 + 1 • Greek numerology considered 6 the most beautiful of all numbers, representing marriage, health and beauty – since it was the sum of its own parts. • 6 represented the goddess of love Venus for it is the product of 2, which represents female, and 3, which represents male. • God created the world in 6 days.

  9. 28 = 14 + 7 + 4 + 2 + 1 • Cycle of moon is 28 days. • Show that 496 and 8128 are the next two perfect numbers.

  10. Perfect Number Characteristics • Are there infinitely many perfect numbers? • Is every perfect number even? • Do all perfect numbers end in 6 or 8? • Is there a formula for generating perfect numbers?

  11. Are there infinitely many perfect numbers? • Not Known – unsolved problem • Only 24 known perfect numbers. • Cataldi (1603 ) - 5th through 7th 33, 550, 336 8, 589, 869, 056 137, 438, 691, 328 • Euler (1772) – 8th perfect number 2, 305, 483, 008, 139, 952, 128

  12. Do all perfect even numbers end in 6 or 8? • Not known if any odd perfect numbers exist. • Proven all perfect number that are even do end in 6 or 8.

  13. Perfect Number Form • An even perfect number must have the form 2p-1(2p-1) where 2p – 1 is prime. • Mersenne Primes – primes of form 2p - 1.

  14. Perfect Numbers related to Mersenne Primes • 6 = 2(22 – 1) • 28 = 22(23-1) • 496 = 24(25-1) • 8128= 26(27-1) • 212(213-1) • 216(217-1) • 218(219-1) • 230(231-1) • Largest Know Perfect Number is 219936(219937-1)

  15. Leopold Kronecker Born: 7 Dec 1823 in Liegnitz, Prussia (now Legnica, Poland)Died: 29 Dec 1891 in Berlin, Germany

  16. Leopold Kronecker(1823-1891) • God made the integers, all the rest is the work of man. • All results of the profoundest mathematical investigation must ultimately be expressed in the simple form of properties of integers.

  17. Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany)Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)

  18. Karl Friedrich Gauss(1777-1855) • Mathematics is the queen of sciences and number theory is the queen of mathematics.

  19. Leonhard Euler Born: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St Petersburg, Russia

  20. Leonard Euler (1707-1783) • There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove – only observations have led us to their knowledge.

  21. Famous Number Theory Conjectures • Goldbach’s Conjecture: Every even number greater than 4 can be expressed as the sum of two odd primes. • Twin Prime Conjectures: There is an infinite number of pairs of primes whose difference is two.

  22. Goldbach’s Conjecture • Examine the first several cases • Easy to understand question • Difficult to prove • 6 = 3 + 3 • 8 = 3 + 5 • 10 = 3 + 7 • 12 =5 + 7 • Try some larger even numbers

  23. Pierre de Fermat Born: 17 Aug 1601 in Beaumont-de- Lomagne, FranceDied: 12 Jan 1665 in Castres, France

  24. Fermat’s Last Theorem There are no non-zero whole numbers a, b, c where a n + b n = c nfor n a whole number greater than 2. • Extension of Pythagorean Theorem a 2 + b 2 = c 2 • Pythagorean triples (3,4,5), (5,12,13) • Try letting n = 3 and finding cases that work. Use the TI-73 to explore.

  25. Divides • If a, b  Z with a  0, then a divides b if there exists a c  Z such that a  c = b. • Notation: a | b a divides b b is a multiple of a a is a divisor of b a | b a does not divide b

  26. Properties Of Divides • a 0 then a | 0 and a | a. • 1 | b for all b  Z. • If a | b then a | b·c for all c  Z. • Transitivity: a | b and b | c implies a | c. • a | b and a | c implies a | (bx+cy), x,y  Z. • a | b and b  0 implies |a| |b|. • a | b and b | a implies a =b

  27. Proof of Divide Property • a | b and a | c implies a | (bx+cy), x,y  Z • Proof:

  28. Divisibility Rules for 2, 5, and10 • Rule for 2: If n is even, then 2 | n • Rule for 5: If n has a ones digit of 0 or 5, then 5 | n. • Rule for 10: If n has a ones digit of 0, then 10 | n.

  29. Proof of Divisibility by 5 • Proof: Let n be an integer ending in 0 or 5. Write n out in expanded notation and verify that 5 divides n.

  30. Divisibility Rules For 3, 6 and 9 • Rule for 3: If 3 divides the sum of the digits of n, then 3 | n. • Rule for 6: If 2 | n and 3 | n, then 6 | n. • Rule for 9: If 9 divides the sum of the digits of n, then 9 | n.

  31. Exploration of Divisibility by 3 • 3 | (2+1+ 6) so 3 | 216. • Expand 216 = 2100 + 110 + 6. • Convert to terms divisible by 3.

  32. Proof of Divisibility by 3 • Proof: Show for 3 digit number, then expand to higher cases

  33. Principle Of Mathematical Induction Let S(n) be a statement involving the integers n. Suppose for some fixed integer no two properties hold: • Basis Step: S(no) is true; • Induction Step: If S(k) is true for k  Z where k no ,then S(k+1) is true. • THEN S(n) is true for all nZ , n  n0

  34. Mistaken Induction? • Prove: an = 1 for any n  Z+ {0}, a  Real, a  0. Proof: Basis Step: a o = 1 so true for n = 0 Induction Step: Suppose for some integer k that a k = 1 then ak+1 = a k a k / ak-1 = (1  1)/1 = 1 By induction an = 1. • What is the error in this argument?

  35. Math Induction ProofDivisibility by 3 • Proof: Let n be any integer such that the sum of its digits is divisible by 3.

  36. Exploration • Use the rule for divisibility by 3 to prove the rules for 6 and 9.

  37. Divisibility Rules for 4, 8 and 12 • Rule for 4: If 4 divides the last 2 digits of n, then 4 | n. • Rule for 8: If 8 divides the last 3 digits of n, then 8 | n. • Rule for 12: If 3 | n and 4 | n, then 12 | n.

  38. Exploration • Parallel the argument for divisibility by 3 to prove divisibility by 4. • Divisibility by 8 and 12 follow from the divisibility by 4.

  39. Divisibility Rules for 7,11 and 13 • Rule for 7: If 7 divides the alternating sum/difference of 3 successive digits then 7 | n. • Rule for 11: If 11 divides the alternating sum/difference of 3 successive digits then 11 | n. • If 13 divides the alternating sum/difference of 3 successive digits, then 13 | n.

  40. Example • Does 7 divide 515, 592? • 592 – 515 = 77 Since 7 | 77, then 7 | 515,592 • Try 1,516,592

  41. Proof of Divisibility by 7 • Proof: Argue for 6 digit number and use Math Induction to verify generalization

  42. Prime Numbers • Prime Number: If p is an integer, p > 1, and p has only 2 positive integer divisors, then p is called a prime number. • Composite Number: If p > 1 and p is not prime, then p is called a composite number.

  43. Fundamental Theorem Of Arithmetic • Every integer n 2 is either prime or can be factored into a product of primes. • Prove requires a stronger form of Mathematical Induction

  44. Strong Principle Of Mathematical Induction • Let S(n) be a statement involving the integer n. Suppose for some fixed integer n0. • Basis Step: S(n0) is true • Induction Step: If S(n0), S(n0+1)…S(k) are true for k Z , k  n0 then S(k+1) is true. • THEN S(n) is true for all integers n  n0

  45. Proof of FTA • Proof: Use Strong Math Induction

  46. Sieve of Eratosthenes • Finds primes up to n from knowledge of primes up to n • Easy to implement in a graphical form

  47. Thank You !!

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