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 1  2

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 = 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 = 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 ) - 5 th through 7 th 33, 550, 336 8, 589, 869, , 438, 691, 328 Euler (1772) – 8 th 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 2 p-1 (2 p -1) where 2 p – 1 is prime. Mersenne Primes – primes of form 2 p - 1.

Perfect Numbers related to Mersenne Primes 6 = 2(2 2 – 1) 28 = 2 2 (2 3 -1) 496 = 2 4 (2 5 -1) 8128= 2 6 (2 7 -1) 2 12 ( ) 2 16 ( ) 2 18 ( ) 2 30 ( ) Largest Know Perfect Number is ( )

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

Leopold Kronecker( ) 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( ) 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 ( ) 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 = = = =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 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.

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 = 2   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 n o two properties hold: Basis Step: S(n o ) is true; Induction Step: If S(k) is true for k  Z where k  n o,then S(k+1) is true. THEN S(n) is true for all n  Z, n  n 0

Mistaken Induction? Prove: a n = 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 a k+1 = a k a k / a k-1 = (1  1)/1 = 1 By induction a n = 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 n 0. Basis Step: S(n 0 ) is true Induction Step: If S(n 0 ), S(n 0 +1)…S(k) are true for k  Z, k  n 0 then S(k+1) is true. THEN S(n) is true for all integers n  n 0

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 !!