MAT310: Number Theory Overview

Abstract:

We've been over a lot of terrain in this course. Some themes have, however, been appearing over and over. Well-ordering, the division algorithm, prime factorizations, Diophantine equations, various proof techniques (including contradiction and induction), and certainly math history have had important roles in this course.

I hope that you have a better sense of where mathematics has come from (and maybe even where it's going!). One thing we've seen is that, in spite of number theory's purity, it is also absolutely essential to some modern applications of mathematics (e.g. information security). Pythagorean triples arose from an application, and then formed the basis of Fermat's last theorem.

Section 1.1

Some useful preliminaries:

• Well-Ordering Principle
• Archimedean property
• First Principle of Finite Induction
• Second Principle of Finite Induction
• Binomial Theorem
• Pascal's rule

Section 2.1

• Division Algorithm: Given integers a and b, with b>0, there exist unique integers q and r satisfying

  

  with . q is called the quotient, and r is called the remainder.

  Corollary: Given integers a and b, with , there exist unique integers q and r satisfying

  

  with .

Section 2.2

• common divisor
• greatest common divisor
• relatively prime:
• Theorem 2.2: For integers a, b, c, the following hold:
  1. a | 0, 1 | a, a | a
  2. a | 1 if and only if
  3. If a | b and c | d, then ac|bd.
  4. If a | b and b | c, then a|c.
  5. a | b and b | a if and only if
  6. If a | b and , then .
  7. If a | b and a | c, then a|(bx+cy) for arbitrary integers x and y.
• Theorem 2.3: Given integers a and b, not both zero, there exists integers x and y such that

  

  Corollary: If a and b are given integers, not both zero, then the set

  

  is precisely the set of all multiples of .

• Theorem 2.4: Let a and b be integers, not both zero. Then a and b are relatively prime if and only if there exist integers x and y such that 1=ax+by.

  Corollary 1: If gcd(a,b)=d, then gcd(a/d,b/d)=1.

  Corollary 2: If a|c and b|c, with gcd(a,b)=1, then ab|c.

• Theorem 2.5 (Euclid's lemma): If a|bc, with gcd(a,b)=1, then a|c.
• Theorem 2.6: Let a and b be integers, not both zero. For a positive integer d, if and only if
  1. d|a and d|b, and
  2. Whenever c|a and c|b, then c|d.

Section 2.3

• least common multiple
• Lemma: If a=qb+r, then
• Euclidean Algorithm:

  

  (i.e. , so the final remainder is 0). Then .

• Theorem 2.7: if k>0, then .

  Corollary: if , then .

• Theorem 2.8: For positive integers a and b

  

  Corollary: For positive integers a and b

  

Section 2.4

• Diophantine equation
• Theorem 2.9: The linear Diophantine equation ax+by=c has a solution iff d|c, where . If is any particular solution of this equation, then all other solutions are given by

  

  for integral values of t.

  Corollary: If , and is any particular solution of the equation ax+by=c, then all other solutions are given by

  

  for integral values of t.

Section 3.1

• prime, composite
• Theorem 3.1: If p is prime and p | ab, then p|a or p|b.

  Corollary 1: If p is prime and , then for some k, .

  Corollary 2: If are all prime and , then for some k, .

• Theorem 3.2 (Fundamental Theorem of Arithmetic): Every positive integer n>1 can be expressed as a product of primes uniquely (up to the order of the primes in the product).

  Corollary: Any positive integer n>1 can be written uniquely in a canonical form

  

  where, for i=1,2,...,r each is a positive integer and each is a prime, with .

• Theorem 3.3 (Pythagoras): is irrational.

Section 3.2

• The sieve of Eratosthenes
• Theorem 3.4 (Euclid): The primes are infinite in number.
• Theorem 3.5: If is the prime, then .

  Corollary: For , there are at least n+1 primes less than .

Section 3.3

• Various famous interesting conundrums, mysteries, conjectures, etc. are discussed, including
  • The Goldbach Conjecture;
  • Twin Primes, and other gaps between primes;
  • Dirichlet's primes of the form a+kb, with gcd(a,b)=1; and
  • Primes of various forms given by the division algorithm.
• Lemma: The product of two or more integers of the form 4n+1 is of the same form.
• Theorem 3.6: There are infinitely many primes of the form 4n+3.
• Theorem 3.7 (Dirichlet): If a and b are relatively prime positive integers, then the arithmetic progression

  

  contains infinitely many primes.

• Theorem 3.8: If all the n>2 terms of the arithmetic progression

  

  are prime numbers, then the common difference d is divisible by every prime q<n.

Section 4.2

• congruence modulo n
• complete set of residues
• Theorem 4.1: For arbitrary integers a and b, if and only if a and b leave the same nonnegative remainder when divided by n.
• Theorem 4.2: Let n>1 be fixed and a, b, c, and d be arbitrary integers. Then the following properties hold:
  1. 
  2. If , then .
  3. If and , then .
  4. If and , then , and .
  5. If , then , and .
  6. If , then for any positive integer k.
• Theorem 4.3: If , then , where .

  Corollary 1: If and , then .

  Corollary 2: If (p prime), and p does not divide c, then .

• Problem #13: If and , then , where . Hence, whenever and are relatively prime, .

Section 4.3

• base b place-value notation
• Theorem: Given any integer b>1, any integer may be written uniquely in base b place-value notation.
• Theorem 4.4: Let be a polynomial function of x with integral coefficients . If , then .

  Corollary: If a is a solution of the congruence , and , then b is also a solution.

• Theorem 4.5/4.6: Let

  

  be the decimal expansion of positive integer N, , and let . Then

  • .
  • Let . Then .

Section 4.4

• linear congruence
• Theorem 4.7: The linear congruence has a solution if and only if d | b, where . If d|b, then the linear congruence has d mutually incongruent solutions modulo n.

  Corollary: If gcd(a,n)=1, then the inear congruence has a unique solution modulo n.

• Theorem 4.8 (The Chinese Remainder Theorem): Let be positive integers such that for . Then the system of linear congruences

  

  has a simultaneous solution which is unique modulo . The unique solution is of the form

  

  where and is the unique solution to the linear congruence .

• Theorem 4.9: The system of linear congruences

  

  has a unique solution whenever .

Section 5.3

• Theorem 5.1 (Fermat's Theorem): Let p be prime and suppose that p does not divide a. Then .

  Corollary: If p is a prime, then for any integer a.

• Lemma: If p and q are distinct primes with and , then .

Section 5.4

• Theorem 5.4 (Wilson's Theorem): If p is prime, then

  

  Converse to Wilson's Theorem): If

  

  then p is prime.

• Theorem 5.5: The quadratic congruence , where p is an odd prime, has a solution if and only if .

Section 6.1

• 

  and

  

• A number-theoretic function is said to be multiplicative if

  

  whenever .

• Theorem 6.1 If is the prime factorization of n>1, then the positive divisors of n are precisely those integers of the form , where for i in .
• Theorem 6.2 If is the prime factorization of n>1, then

  1. 

     and

  2. 

• 

  and

  

• Theorem 6.3 The functions and are multiplicative functions.
• Lemma If , then the set of positive divisors of mn consists of all products , where , , and ; furthermore these products are all distinct.

  Theorem 6.4 If f is a multiplicative function and F is defined by

  

  then F is also multiplicative.

Section 7.2

• Euler's : For , let denote the number of positive integers not exceeding n that are relatively prime to n.
• Theorem 7.1: If p is prime and k>0, then

  

• Lemma: Given integers a, b, c, if and only if and .

  Theorem 7.2: The function is a multiplicative function.

• Theorem 7.3: If the integer n>1 has the prime factorization , then

  

• Theorem 7.4: For n>2, is an even integer.

Section 7.3

• Theorem 7.5 (Euler): If and , then .

  Corollary: Fermat's Little theorem

Section 7.5

• Cipher - the code
• Plaintext - the message to be encrypted
• Ciphertext - the encrypted message
• Frequency Analysis - using the known distribution of letters (or words) to break a code.
• Caesar Cypher (circa 50 B.C.)
• Vigenère Cypher (1586)
• Hill's cipher (1929)

Section 7.5b

• The RSA algorithm

Section 10.2

• perfect number
• Mersenne number: , with . If is prime, then it's called a Mersenne prime
• Theorem 10.1: If is prime (k>1), then is perfect, and every even perfect number is of this form.
• The test of a perfect number is if

  

Section 11.1

• Pythagorean triple
• Lemma 1: If x, y, z is a primitive Pythagorean triple, then one of the integers x or y is even, while the other is odd.

  Lemma 2: If , where gcd(a,b)=1, then a and b are powers. That is, there exist positive integers and for which and .

  Theorem 11.1: All solutions of the Pythagorean equation

  

  satisfying the conditions

  

  are given by the formulas

  

  For integers s>t>0 such that gcd(s,t)=1 and .

Section 11.2

• Theorem 11.3: The Diophantine equation has no solution in the positive integers x, y, and z.

  Corollary: The equation has no solution in the positive integers x, y, and z.

  Corollary: The equation has no solution in the positive integers x, y, and z.

• Theorem 11.4: The Diophantine equation has no solution in the positive integers x, y, and z.

Section 13.1

• Fibonacci numbers:

  

  for , where .

• Theorem 13.1: For the Fibonacci sequence, for every .
• Theorem 13.2: For and , .
• Lemma: If m = qn+r, then

  Theorem 13.3: The greatest common divisor of two Fibonacci numbers is again a Fibonacci number; specifically where .

  Corollary: In the Fibonacci sequence, if and only if m | n for .

Section 13.2

• Theorem 13.4: Any positive integer N can be expressed as a sum of distinct Fibonacci numbers, no two of which are consecutive; that is,

  

  where and for (the Zeckendorf representation).

•