22 sample documents related to MATH 413

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  980930.pdf

  UMBC MATH 413
  

  Theorem 0.1. If gcdm; n = 1 then Cm Cn Cmn = 6= If gcdm; n = r = 1 then Cm Cn C mn Cr Cmn 6 = r We demonstrate this with two examples - C C C and C C C . 6= = The group C C is generated by the single element 1; 1, which has order 6. We see
   
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  980916.pdf

  UMBC MATH 413
  

  1 1.1 Unique Factorization The Counterexample Revisited Recall my favorite counterexample: Z[ -5] where we define a norm (absolute value) as | a + b -5 |= a2 + 5b2 . We analyze it with a little computation. First write down a list of possible val
   
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  981007.pdf

  UMBC MATH 413
  

  0.1 Di e Hellman Key Agreement Algorithm Compute 232 mod 53 e ciently. Recall the Russian Peasant algorithm. Now solve the equation 2x 6 mod 53 for x. This is a much more di cult problem. De nition 0.1. The Discrete Logarithm Problem also called th
   
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  980914.pdf

  UMBC MATH 413
  

  1 1.1 1.1.1 The Euclidean Algorithm Examples & Counterexamples Polynomials Consider how we can build a division algorithm for the polynomials with rational coecients, Q[x]. First we need a notion of size of the polynomial - for this we use the degr
   
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  980928.pdf

  UMBC MATH 413
  

  1 1.1 The Ring of Integers mod N The Group of Units Denition 1.1. A unit in a ring R is an element which has an inverse. Theorem 1.1. In Zm , [a] is a unit i gcd(a, m) = 1. Denition 1.2. (N ) is the number of units in ZN . (This is sometimes call
   
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  MidTerm_Solns.pdf

  UMBC MATH 413
  

  MidTerm Solns Math 413, Spring 2009 Mar, 2009 Open Book Calculators & Laptops Allowed (but brute force solutions not expected) Show all work 1. (40 pts) (a) Compute (37) and (37). Ans: As 37 is prime (37) = 37 - 1 = 36 and (37) = 37 - 1 = 36. (b) C
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  What is Number Theory? Number Theory is the study of integers and rational numbers and algebraic extensions of the rationals and similar stuff. Find all (integer/rational) solutions to: a2 + b 2 = c 2 (Pythagorean Triples) a3 + b 3 = c 3 (Fermat\'s La
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  1. Divisibility, Primes & Congruences Math 413: Number Theory Robert Campbell UMBC February 12, 2009 Robert Campbell (UMBC) 1. Divisibility February 12, 2009 1 / 55 Divisibility Def: a divides b (denoted a|b) if there exists an integer x such
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  2. Group of Units Math 413: Number Theory Robert Campbell UMBC March 20, 2009 Robert Campbell (UMBC) 2. Group of Units March 20, 2009 1/1 Equivalence Relations Def: An equivalence relation, on a set S is a subset E of S S with the properties
   
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  hw4.pdf

  UMBC MATH 413
  

  HW4: , and Primitive Elements Robert Campbell Math 413, Spring 2009 1. (a) List the elements of Z and compute (21) 21 (b) Compute the orders of the elements of Z and compute (21) 21 (c) Compute (54) and (54) (d) Compute (5000) and (5000) 2. Find all
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  4. Quadratic Residues & Equations Math 413: Number Theory Robert Campbell UMBC April 30, 2009 Robert Campbell (UMBC) 4. Quadratics April 30, 2009 1 / 49 Quadratic Residues Is every element of Zp a square? Examples: p = 13 0 0 1 1 2 4 3 9 4 3
   
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  MidTerm.pdf

  UMBC MATH 413
  

  MidTerm Math 413, Spring 2009 Mar, 2009 Open Book Calculators & Laptops Allowed (but brute force solutions not expected) Show all work 1. (40 pts) (a) Compute (37) and (37). (b) Compute (72) and (72). (c) Is 2 primitive (mod 37)? Is 3 primitive (mo
   
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  hw5.pdf

  UMBC MATH 413
  

  HW5: Primitive Elements & Equations 1. Assume that {2, 3, 5, 7} are prime. Using the Lucas-Kraitchik primality test (recursively prove primality by finding primitive elements): (a) Prove that 31 is prime. (b) Prove that 157 is prime. 2. Find all sol
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  4. Analytic Number Theory Math 413: Number Theory Robert Campbell UMBC April 7, 2009 Robert Campbell (UMBC) 3. Analytics April 7, 2009 1 / 32 Analytic Number Theory Def: Analytic number theory is the application of methods from analysis (calc
   
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  hw6.pdf

  UMBC MATH 413
  

  HW6: Finite Fields & Quadratic Residues Robert Campbell Math 413, Spring 2009 1. (a) Compute the Jacobi symbol ( 3350 ) 3337 (b) Is 3350 a square mod 3337? If so, compute 3350(mod 3337). 2. For each of the following integers, determine for which
   
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  slides_handouts.pdf

  UMBC MATH 413
  

  5. Continued Fractions Approximating Fractions Example: Apply the Extended
   
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  hw7.pdf

  UMBC MATH 413
  

  HW7: Analytic, Continued Fractions & Sums of Squares Robert Campbell Math 413, Spring 2009 1. Show that if (n) is odd then n is a square or twice a square. 2. Compute continued fraction expansions of: (a) 101/73 (b) The cube root of 3 (the first 7 p
   
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  QuadNumbFlds_notes.pdf

  UMBC MATH 413
  

   
   
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  EllipticCurves.pdf

  UMBC MATH 413
  

   
   
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  hw8.pdf

  UMBC MATH 413
  

  HW8: Number Fields & Elliptic Curves Robert Campbell Math 413, Spring 2009 1. Consider Z[ 11], the ring of integers in Q[ 11] Find the minimal polynomial of the element 2 + 3 11 Find all the units in the ring Z[ 11] 2. Consider Z[ -13], the r
   
• 
  LucasPrimTest.txt

  UMBC MATH 413
  

  Lucas Primality Test Examples Lucas-Kraichik-Lehmer-Selfridge Test Edouard Lucas (1876 & 1891) Example: Prove 1237 is prime: Proof: 1237-1 factors as 2, 2, 3, 103 Check whether 2 is primitive element 1: 2^(1237-1) (mod 1237) = 1 2: 2^(1237-1)/2) = 2^6
   
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  slides.pdf

  UMBC MATH 413
  

  2. Group of Units Math 413: Number Theory Robert Campbell UMBC March 20, 2009 Robert Campbell (UMBC) 2. Group of Units March 20, 2009 1/1 Equivalence Relations Def: An equivalence relation, on a set S is a subset E of S S with the properties: Reflexitivit