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,

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.

• 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 = 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 KroneckerBorn: 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 EulerBorn: 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 FermatBorn: 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 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 = 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 = 0Induction Step: Suppose for some integer kthat a k = 1 then

ak+1 = a k a k / ak-1 = (1 1)/1 = 1By 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

Thank You !!