Number Theory

1 / 50

# Number Theory - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Number Theory

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

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.

• 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.
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, SwitzerlandDied: 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, FranceDied: 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 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.
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100 + 110 + 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 nZ , 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 ProofDivisibility 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