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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. • Deductive Reasoning: Patterns are formalized and verified with proof. Pythagoras of Samos Born: about 569 BC in Samos, Ionia Died: about 475 BC 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. • 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. • Discovery of 2 as irrational number 1 2 1 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. • 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. • 28 = 14 + 7 + 4 + 2 + 1 • Cycle of moon is 28 days. • Show that 496 and 8128 are the next two perfect numbers. 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? 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 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. 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. 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) Leopold Kronecker Born: 7 Dec 1823 in Liegnitz, Prussia (now Legnica, Poland) Died: 29 Dec 1891 in Berlin, Germany 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. Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died: 23 Feb 1855 in Göttingen, Hanover (now Germany) Karl Friedrich Gauss(1777-1855) • Mathematics is the queen of sciences and number theory is the queen of mathematics. Leonhard Euler Born: 15 April 1707 in Basel, Switzerland Died: 18 Sept 1783 in St Petersburg, Russia 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. 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. 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 Pierre de Fermat Born: 17 Aug 1601 in Beaumont-de- Lomagne, France Died: 12 Jan 1665 in Castres, France Fermat’s Last Theorem There are no non-zero whole numbers a, b, c where a n + b n = c n for n a whole number greater than 2. • Extension of Pythagorean Theorem a2+b2 =c2 • Pythagorean triples (3,4,5), (5,12,13) • Try letting n = 3 and finding cases that work. Use the TI-73 to explore. 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 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 Proof of Divide Property • a | b and a | c implies a | (bx+cy), x,y Z • Proof: 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. 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. 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. Exploration of Divisibility by 3 • 3 | (2+1+ 6) so 3 | 216. • Expand 216 = 2100 + 110 + 6. • Convert to terms divisible by 3. Proof of Divisibility by 3 • Proof: Show for 3 digit number, then expand to higher cases 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 nZ , n n0 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? Math Induction Proof Divisibility by 3 • Proof: Let n be any integer such that the sum of its digits is divisible by 3. Exploration • Use the rule for divisibility by 3 to prove the rules for 6 and 9. 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. Exploration • Parallel the argument for divisibility by 3 to prove divisibility by 4. • Divisibility by 8 and 12 follow from the divisibility by 4. 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. Example • Does 7 divide 515, 592? • 592 – 515 = 77 Since 7 | 77, then 7 | 515,592 • Try 1,516,592 Proof of Divisibility by 7 • Proof: Argue for 6 digit number and use Math Induction to verify generalization 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. 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 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 Proof of FTA • Proof: Use Strong Math Induction Sieve of Eratosthenes • Finds primes up to n from knowledge of primes up to n • Easy to implement in a graphical form Thank You !!