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## Number Theory

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### Number Theory

### Pythagoras of Samos

### Leopold Kronecker

### Johann Carl Friedrich Gauss

### Leonhard Euler

### Pierre de Fermat

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.

Born: about 569 BC in Samos, IoniaDied: 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.

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)

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.

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.

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

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 = 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 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.

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

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