Number Theory and Methods of Proof

1
Number Theory and Methods of Proof

• Content Properties of integer, rational
• and real numbers.
• Underlying theme Methods of
  mathematical proofs.

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This Lecture

• Even and odd integers
• Prime and composite integers
• Constructive and nonconstructive proofs
• Method of direct proof

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Even and Odd Integers

• Definition An integer n
• ? is even iff
• ? an integer k such that n2k
• ? is odd iff
• ? an integer k such that n2k1.
• Ex If x and y are integers,
• is 4x2-2xy7 even or odd?

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Prime and Composite Integers

• Definition An integer n (which is gt1)
• ? is prime iff ? positive integers r and s,
• if nrs then r1 or s1
• ? is composite iff ? positive integers r and s
  such that nrs and r?1 and s?1.
• Examples 5, 7, 23 are prime
• 4, 22, 16042 are composite.

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Proving Existential Statements

• Existential statement ?x?D such that P(x)
• 2 proof methods for existential statements
• ? Constructive proofs
• ? Nonconstructive proofs.

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Constructive Proofs of Existence

• 2 ways
• 1) Find x that makes P(x) true.
• 2) Give a set of directions for finding such x.
• Example
• 1) There are integer numbers a,b and c
• such that a2b2c2.
• Proof For a3, b4 and c5, 324252.

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Constructive Proofs of Existence

• 2) Suppose a,b ?Z such that 1ltaltb.
• Prove that there is a composite even integer
  c
• such that a2ltcltb2 .
• Proof By division into cases
• (a) Suppose a is even.
• Then a2k for some integer k. (by
  definition)
• Hence cab (2k)b 2(kb) is even
  integer (because kb is an
  integer)
• cab is composite (because a?1 and b?1)
• cabgta2 (because altb).
• cabltb2 (because altb).
• (b) Suppose b is even.
• (c) Suppose both a and b are odd.

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Nonconstructive Proofs of Existence

• 2 ways
• (1) Show that the existence of x
• is guaranteed by an axiom
• or a previously proved theorem.
• (2) Show that the assumption that there
• is no such x leads to a contradiction.
• Disadvantage Often these methods give
• no clue how to find x.

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Proving Universal Statements

• Universal statement ?x?D if P(x) then Q(x)
• Proof methods for universal statements
• ? Method of exhaustion
• ? Method of generalizing from the
  generic particular.
• (show the property for a particular but
• arbitrarily chosen x)

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Method of Direct Proof

• The statement ?x?D if P(x) then Q(x).
• Suppose x is a particular but arbitrarily
  chosen element of D
• for which P(x) is true
• Show the conclusion Q(x) is true by using
• ? definitions
• ? previously established results
• ? rules of logical inference.

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Method of Direct Proof (Ex.)

• Show ?x?Z if x is odd
• then 3x9 is even.
• Proof Suppose x is an arbitrarily chosen odd
  integer.
• Then x2k1 for some integer k. (by
  definition)
• So 3x9 3(2k1)9 (by substitution)
• 6k39 (by distributive law)
• 2(3k6) (by factoring out a 2)
  ()
• 3k6 is an integer. ()
• Hence 3x9 is even
• based on (), () and definition of even
  integers.
• (this is what we needed to show)

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Directions for writing proofs

• Write the theorem to be proved.
• Clearly mark the beginning of your proof
• with the word Proof.
• 3) Make your proof self-contained.
• (Identify all variables used in the proof
• state the sources of outside facts).
• 4) Write proofs in complete English
  sentences.