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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.

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number theory

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.

slide4
Inductive Reasoning: Patterns are discovered and generalized.
  • Deductive Reasoning: Patterns are formalized and verified with proof.
pythagoras of samos

Pythagoras of Samos

Born: about 569 BC in Samos, IoniaDied: about 475 BC

pythagorean society
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.

slide7
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
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.
slide10
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.
slide11
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
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?
slide13
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

slide14
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
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
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

Leopold Kronecker

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

leopold kronecker 1823 1891
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

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
Karl Friedrich Gauss(1777-1855)
  • Mathematics is the queen of sciences and number theory is the queen of mathematics.
leonhard euler

Leonhard Euler

Born: 15 April 1707 in Basel, SwitzerlandDied: 18 Sept 1783 in St Petersburg, Russia

leonard euler 1707 1783
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
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
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

Pierre de Fermat

Born: 17 Aug 1601 in Beaumont-de- Lomagne, FranceDied: 12 Jan 1665 in Castres, France

fermat s last theorem
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
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
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
Proof of Divide Property
  • a | b and a | c implies a | (bx+cy), x,y  Z
  • Proof:
divisibility rules for 2 5 and10
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 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
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
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 of Divisibility by 3
  • Proof: Show for 3 digit number, then expand to higher cases
principle of mathematical induction
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
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
Math Induction ProofDivisibility by 3
  • Proof: Let n be any integer such that the sum of its digits is divisible by 3.
exploration
Exploration
  • Use the rule for divisibility by 3 to prove the rules for 6 and 9.
divisibility rules for 4 8 and 12
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.
exploration40
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
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
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 of Divisibility by 7
  • Proof: Argue for 6 digit number and use Math Induction to verify generalization
prime numbers
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
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
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 of FTA
  • Proof: Use Strong Math Induction
sieve of eratosthenes
Sieve of Eratosthenes
  • Finds primes up to n from knowledge of primes up to n
  • Easy to implement in a graphical form