9+Number+Theory

choose from Number Theory or Game Theory - study for 3 weeks.

__**Number Theory rough daily plan**__

__**3-4 Days of Fun Puzzles**:__ Including introductory ideas of what Number Theory deals with - whole numbers, their factors, primes, etc.

Palindrome Game - EXPLORATION: Palindrome Game (check MFT Algebra Notes as well) - 2 digits, try 3 digits Send More Money Puzzles from IMP 432,432 is a multiple of 1001, which is 13x11x7 (look up in H&P) Tricks with multiplying, e.g. digits of multiples of 9 add up to 9, for multiples of 11 if sum of odd digits - sum of even digits = multiple of 11 or 0, and more Pythagorean Triples - how to generate with m, n (a=diff of squares, b=2mn, c=sum of squares) Perfect and Amicable numbers -deficient & abundant Perfect Numbers: Collatz Conjecture Words game - each letter of alphabet is assigned a number, can you find words with ...
 * look at PROMYS PSET 1 # Q3 - summation of 1/divisors of N = 2 for perfect numbers exploration
 * PSET 1 #12, 13

__**3-4 Days of Clock/Mod stuff**__ Do Eggs/Remainder problems Clock Arithmetic: Questions like, what is 1/2 in Z7 : 2x = 1 what is x. PROMYS (PSET 0 # P3, P4) (PSET 1 # P11) Also different bases - why powers of 3 in base 3 are nice, but in base 2, they aren't, or look at repeating and nonrepeating fractions in base 10. (PSET 1 # 10) What is a unit? - PROMYS Lecture 4, PSET 2 # 9 - are things cyclic?, generators. . .PSET 3 # 9 Find all the perfect squares in a certain mod (PROMYS Notes Day 8) Multiplicative inverses => given a certain mod, the numbers that are relatively prime to the base will have multiplicative inverses

Clock Arithmetic - describe how its used, http://www.shodor.org/interactivate/lessons/ClockArithmetic/

also can play with ciphers as an application
 * http://www.shodor.org/interactivate/activities/CaesarCipher/
 * http://www.shodor.org/interactivate/activities/CaesarCipherTwo/
 * http://www.shodor.org/interactivate/activities/CaesarCipherThree/

Pascal's Triangle values using modular arithmetic creates smaller designs:[[@http://Think%20about%20prove,%20disprove%20and%20salvages%20if%20possible%20-%20simple%20questions%20like%20a%7Ca%20for%20all%20a%20or%20a%7Cb%20-%3E%20b%7Ca%20%20%20%20%20%20Number%20Systems%20and%20Rules%20of%20Arithmetic:%20%20%20%20%20%20Closure%20of%20addition%20%20%20%20%20Commutativity%20of%20addition%20%20%20%20%20Associativity%20of%20addition%20%20%20%20%20Additive%20identity%20%20%20%20%20Additive%20Inverses%20%20%20%20%20%20Closure%20of%20multiplication%20%20%20%20%20Commutativity%20of%20multiplication%20%20%20%20%20Associativity%20of%20multiplication%20%20%20%20%20Multiplicative%20identity%20%20%20%20%20Multiplicative%20inverse%20%28key%20distinguisher%20between%20number%20systems%29%20%20%20%20%20%20Distribution%20over%20multiplication%20and%20addition%20%20%20%20%20%20Rings%20vs.%20Fields%20head%20into%20clock%20arithmetic%20to%20work%20with%20the%20rules%20from%20above.%20%28check%20out%20Z5,%20and%20Rational%20Numbers,%20Q%29%20%20All%20Zn%20where%20n%20is%20prime%20are%20fields%21%20%20Every%20field%20is%20a%20ring%20but%20not%20every%20ring%20is%20a%20field.%20%20A%20field%20is%20a%20ring%20with%20a%20few%20more%20properties%20like%20commutativity%20and%20fields%20have%20multiplicative%20inverses%20that%20are%20well-defined%20for%20each%20non-additive-zero%20element%20%28meaning%20fields%20have%20division%29.%20%20Compare%20and%20contrast%20the%20mathematical%20systems%20Z,%20Q%20,%20R,%202Z,%20Z3,%20Z6,%20Z8,%20Z11%20:%20questions:%20do%20you%20get%20answers,%20do%20you%20get%20unique%20answers,%20closure,%20number%20of%20elements,%20and%20inverses%20and%20identities.%20.%20.| http://www.shodor.org/interactivate/activities/ColoringRemainder/]]

Think about prove, disprove and salvages if possible - simple questions like a|a for all a or a|b -> b|a
 * 1) Number Systems and Rules of Arithmetic:
 * Closure of addition
 * Commutativity of addition
 * Associativity of addition
 * Additive identity
 * Additive Inverses


 * Closure of multiplication
 * Commutativity of multiplication
 * Associativity of multiplication
 * Multiplicative identity
 * Multiplicative inverse (key distinguisher between number systems)


 * Distribution over multiplication and addition

All Zn where n is prime are fields! Every field is a ring but not every ring is a field. A field is a ring with a few more properties like commutativity and fields have multiplicative inverses that are well-defined for each non-additive-zero element (meaning fields have division). Compare and contrast the mathematical systems Z, Q, R, 2Z, Z3, Z6, Z8, Z11 : questions: do you get answers, do you get unique answers, closure, number of elements, and inverses and identities. ..
 * 1) Rings vs. Fields head into clock arithmetic to work with the rules from above. (check out Z5, and Rational Numbers, Q)


 * 1) Ring Proofs:
 * Additive identity is unique
 * ax0=0 for any a that is an element of the ring

__**3-4 Days of Primes**__ T or F? All numbers can be written as a product of primes

Is there a way to know how many different factors there are for a number if you know the prime factorization? (play around)

Sieve of Eratosthenes for prime numbers number with an odd number of divisors are perfect squares prime number have no divisors other than 1 and itself discuss why is not a prime b/c of unique prime factorization: 2*3 vs. 1*2*3 vs. 1*1*2*3 Develop x/4 remainder 1 = prime written as sum of 2 square 4k-1 vs. 4k+1 primes Find the smallest number with 7 divisors Suppose p1^(a1)p2^(a2)...pn^(an) = m, then m has exactly (a1 +1)(a2+1). . .(an+1) divisors
 * 1) Activity: Find number with the given number of divisors.
 * Proof of an infinite number of primes
 * Activity: Which primes can be written as a sum of two square numbers?
 * Back to divisor game: prime factorization -> rotations through the exponents give all the divisors
 * Proof: All numbers can be written as a product of primes
 * Greatest Common Divisor (Euclidean Algorithm) - solving equations if GCD(24,9)=3, then 3|(24x + 9y), ie. Diophantine Equations (PROMYS Notes Day 7)

(MAYBE skip groups)
 * __Number Theory Topics not yet assigned:__**
 * 1) EXPLORATION: Symmetries of an equilateral triangle - define a "group" -> ordered ring
 * 2) Group is a finite or infinite set of elements together with a binary operation that satisfy:
 * Closure
 * not necessarily commutative
 * Associativity
 * Identity
 * Inverses


 * 1) Division Algorithm (WOP: well-ordering principle)
 * Define a divisible by b with proof


 * 1) Solving equation x^2 -4 = 0 vs. x^2 + 4 = 0, introducing number system Z[i], Gaussian Integers
 * units of Gaussian Integers (1, -1, i, -i), - (PROMYS Lecture Day 13)
 * which numbers are prime in Z[i] -> all primes of the form 4k - 1 stay prime in Z[i]
 * consider Z[sqrt -5] -> does not have unique factor into prime

Faraday Sequence Conjectures (PSET 2 #2) - lecture Day 5 Congruence Equations x = a and x^2 = a, more advanced students create logarithm table Pset 2 #11 Solving quadratics PROMYS Pset 2 # 1 Chinese Remainder Theorem (Promys lecture 14)

__OTHER RESOURCES__ Resources and handouts: http://www.math.sc.edu/~sumner/numbertheory/mainpage/math580.html Number Course for HS students: (betty look through this to get a clue) http://home.comcast.net/~dgarlock/Index.htm Table of Contents

Friendly Introduction to Number Theory - sent for evaluation copy