Elementary Number Theory

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Elementary Number Theory
Unless otherwise specified, the domain of any
predicate will be the set of natural numbers
N 0, 1, 2, 3, The non-logical
symbols that we can use will be , ,
gt, , 0, 1
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Basic Definitions

• the number a divides b iff
• ? c (b a c)
• the notation is a b
• a number n is even iff it is divisible by 2
• ? k (n 2 k)
• a number n is odd iff
• ? k (n 2 k 1)
• a number p is prime iff
• (p gt 1) ? ?a,b (a b p ? a 1 ? b 1)

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Translating English into 1st order logic

• Examples
• n is the sum of two squares
• ?a ?b (n aa bb)
• there are infinitely many pairs of twin primes
• ? n ?p ( pgtn ? p is prime ? p2 is
  prime)
• Fermats last theorem for any integer n gt 2,
  there is no
• positive integer solution to the equation x n
  y n z n
• ?ngt2 ?? xgt0 ? ygt0 ? zgt0 (x n y n
  z n)
• c is the GCD of a and b
• ?s ?t (a cs ? b ct) ?
• ?d ?s ?t (a ds
  ? b dt) ? d ? c

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Basic types of proofs

• Constructive proof of existential statements
• a concrete example is constructed
• Example prove that the equation x2 y2 z2
  has positive integer solutions
• Proof Consider the integers 3, 4, and 5.
• We see that 32 42 25 52 ,
  therefore the
• statement is true.
  Q.E.D.
• Q.E.D. means quod erat demonstrandum in Latin,
• in English this means this is what
  needed to be
• shown

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Example Prove that there are distinct integers a
and b such that ab ba Proof
Consider the integers 2, and 4. We
see that 24 16 and 42 16, therefore the
statement is true.
Q.E.D. Example Prove
that there are integers a and b such that
7a 12b GCD(7, 12) Proof
Consider

Q.E.D.
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• Non-constructive prove of existence
• this is usually accomplished by a cardinality
  argument,
• namely to show that A?B, we can first
  establish that
• A ? B and then show that B actually has more
  elements
• than A.

Example Prove that there is a non-algebraic real
number. (a non-algebraic number
is also called a
transcendental number.) Definition A number
(real or complex) is said to be algebraic if it
is a solution of a polynomial equation with
integer coefficients. eg. ?2 is a solution to
the equation x2 2 0 (1 ?5)?2 is a
solution to the equation x2 x 1 0
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Clearly every rational number is algebraic. The
following picture summarizes the situation.
We can show that there are only countably many
real algebraic numbers but there are uncountably
many real numbers, therefore there must be some
(in fact quite a lot) of transcendental numbers.
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Even though we knew that there are infinitely
many transcendental numbers, it took
mathematicians many years to actually find an
example
Liouville's number (1844)
0.110001000000000000000001000 ...
which has a one in the 1st, 2nd, 6th, 24th,
etc. places and zeros elsewhere.
Chapernowne's number,
0.12345678910111213141516171819202122232425...
This is constructed by concatenating the
digits of the positive integers. (Can you see the
pattern?) Note e and ? are shown to be
transcendental much later.
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Disproving Universal Statements by Counterexample
To disprove a statement of the form
? x?D, if P(x) then Q(x) find a value of
x in D such that Q(x) is false. Such an x is
called a counterexample.
Example Statement for every real numbers a and
b, if a2 b2, then a b. Disproof Since
(-2)2 4 and 22 4, but -2 ? 2, therefore the
above statement is false.
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• Method of exhaustion for universal statements
• works only for a finite domain
• need to check every single element in the domain
• Example For every 3-digit number n, if the sum
  of the first and
• last digit equals to the middle
  digit, then n is divisible
• by 11.
• Proof n 110 is divisible by 11,
• n 121 is divisible by 11,
• n 132 is divisible by 11,
• n 220 is divisible by 11,
• n 231 is divisible by 11,
• n 242 is divisible by 11,
• n 990 is divisible by 11.
  Q.E.D.

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Method of exhaustion for universal statements
This method fails for infinite domains. However,
quite often we would still like to use brute
force to check the validity of the statement for
a finite subset of the domain. The experience we
get from there will be very valuable to the
construction a real proof. Example Goldbachs
conjecture (1742) Every even
integer n gt 2 is a sum of two
primes. Verify all even numbers n ? 30 . 4 2
2, 6 3 3, 8 3 5, 10 3 7,
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Goldbachs conjecture (1742 )
Up to June 1999, all even numbers up to 41014
have been checked and they all can be written as
a sum of two primes. However, no body can prove
this conjecture yet. The best result so far is
(Chen 1973) Every sufficiently large even number
is the sum of an odd prime and an odd number with
at most two prime factors. Note it is also
shown that every sufficiently large odd number is
a sum of three primes. The lower bound is 107194
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The most controversial application of the
exhaustion method. Four Color Theorem Any
map on the plane can be painted by 4 colors so
that no two adjacent regions have the same
color. The domain of this problem is clearly
infinite, but mathematicians managed to classify
the infinite set of maps into around 1500
different types. After that, they used 1200 hours
of computer time to check all these different
types of maps and concluded that this conjecture
is really correct. For a detail description,
check www-groups.dcs.st-and.ac.uk/history/HistTo
pics/The_four_colour_theorem.html
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• Method of Generalizing from the generic
  particular.
• the most powerful and commonly used method to
  prove an universal statement.
• An object is said to be a generic particular if
  it is randomly chosen from the domain but shares
  the characteristics common to all elements in the
  domain.
• Example Prove that the product of an even number
  and an
• odd number is even.

Example Prove that any odd prime can be written
as the difference of two squares in one and only
one way.
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The power function y x n can be expressed in
terms of , and logical symbols.
where ? (c, a, k) is the Gödel ? -function
defined by
clearly
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Chinese Remainder Theorem
Let m1, m2, m3, mk be pairwise relatively prime
natural numbers. Then the system of congruence
x ? a1 (mod m1)
x ? a2 (mod m2)
..
x ? ak (mod mk) always has a
solution.
Corollary Given any finite sequence of natural
numbers r1, r2, , rn, there are natural numbers
c and a such that for
every i from 1 to n, ? (c, a, i) ri
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proof Let s maxn, r1, r2, , rn 1, and a
s! Then ai 1 gt ri for all i and a 1, 2a 1,
, na 1 are pairwise relatively prime, hence c
exists by the Chinese remainder theorem.