More Number Theory Proofs

1
More Number Theory Proofs

• Rosen 1.5, 3.1

2
Prove or Disprove

• If m and n are even integers, then mn is
  divisible by 4.
• The sum of two odd integers is odd.
• The sum of two odd integers is even.
• If n is a positive integer, then n is even iff
  3n28 is even.
• n2 n 1 is a prime number whenever n is a
  positive integer.
• n2 n 1 is a prime number whenever n is a
  prime number.
• x y ? x y when x,y ? R.
• ?3 is irrational.

3
If m and n are even integers, then mn is
divisible by 4.

• Proof
• m and n are even means that there exists integers
  a and b such that m 2a and n 2b
• Therefore mn 4ab. Since ab is an integer, mn
  is 4 times an integer so it is divisible by 4.

4
The sum of two odd integers is odd.

• This is false. A counter example is 13 4

5
The sum of two odd integers is even.

• Proof
• If m and n are odd integers then there exists
  integers a,b such that m 2a1 and n 2b1.
• m n 2a12b1 2(ab1). Since (ab1) is
  an integer, mn must be even.

6
If n is a positive integer, then n is even iff
3n28 is even.

• Proof We must show that n is even ? 3n28 is
  even, and that 3n28 is even ? n is even.
• First we will show if n is even, then 3n28 is
  even.
• n even means there exists integer a such that n
  2a. Then 3n28 3(2a)2 8 12a2 8 2(6a2
  4) which is even since (6a2 4) is an integer.

7
If n is a positive integer, then n is even iff
3n28 is even (cont.).

• Now we will show if 3n28 is even, then n is even
  using the contrapositive (indirect proof).
• Assume that n is odd, then we will show that
  3n28 is odd. n odd means that there exists
  integer a such that n 2a1.
• 3n28 3(2a1)2 8 3(4a2 4a 1) 8
  2(6a2 6a 5) 1, which is odd.
• Therefore, by the contrapositive if 3n28 is
  even, then n is even.

8
n2 n 1 is a prime number whenever n is a
positive integer.

• Try some examples
• n 1, 111 3 is prime
• n 2, 421 7 is prime
• n 3, 931 13 is prime
• n 4, 1641 21 is not prime and is a counter
  example.
• Not true.

9
n2 n 1 is a prime number whenever n is a
prime number.

• Try some examples
• n 1, 111 3 is prime
• n 2, 421 7 is prime
• n 3, 931 13 is prime
• n 5, 2551 31 is prime
• n 7, 4971 57 is not prime (193).
• Not true.

10
Prove x y ? x y when x,y ? R.

• Note z is equal to z if z?0 and equal to -z if
  z lt 0
• There are four cases
• x y
• ?0 ?0
• lt0 ?0
• ?0 lt0
• lt0 lt0

11
Prove x y ? x y when x,y ? R.

• Case 1
• x,y are both ?0
• Then
• x y x y xy since both x and y
  are positive.

12
Prove x y ? x y when x,y ? R.

• Case 2 x lt 0 and y ?0 so x y -x y
• If y ? -x, then xy is nonnegative and xy
  xy
• Since x is negative, -x gt x, so that
• x y -x y gt xy xy
• If y lt -x, then xy -(xy) -x -y.
• Since y ?0, then y ? -y, so that
• x y -x y ? -x -y xy

13
Prove x y ? x y when x,y ? R.

• Case 3 x?0 and y lt0 so x y x -y
• If x ? -y, then xy is nonnegative and xy
  xy
• Since y is negative, -y gt y, so that
• x y x -y gt xy xy
• If x lt -y, then xy -(xy) -x -y.
• Since x?0, then x?-x , so that
• x y x -y ? -x -y xy

14
Prove x y ? x y when x,y ? R.

• Case 4 x,y are both lt 0
• Then x y -x - y -(xy) xy
• Therefore the theorem is true. This is know in
  mathematics as the Lipschitz condition.

15
Prove that 3?3 is irrational.

• Proof (by contradiction) Assume that 3?3 is
  rational, i.e. that 3?3 a/b for a,b?Z and b?0.
  Since any fraction can be reduced until there are
  no common factors in the numerator and
  denominator, we can further assume that a and b
  have no common factors.

16
Prove that 3?3 is irrational. (cont.)

• Then 3 a3/b3 which means that 3b3 a3 so a3 is
  divisible by 3.
• Lemma When m is a positive integer, then if m3
  is divisible by 3, then m is divisible by 3.
  (Left as an exercise).
• By the lemma since a3 is divisible by 3, then a
  is divisible by 3. Thus ?k?Z ? a 3k.

17
Prove that 3?3 is irrational. (cont.)

• Now, we will show that b is divisible by 3.
• From before, a3/b3 3 ? 3b3 a3 (3k)3.
• Dividing by 3 gives b3 9k3 3(3k3). Therefore
  b3 is divisible by 3 and from the Lemma , b is
  divisible by 3.

18
Prove that 3?3 is irrational. (cont.)

• But, if a is divisible by 3 and b is divisible by
  3, then they have a common factor of 3. This
  contradicts our assumption that our a/b has been
  reduced to have no common factors.
• Therefore 3?3 ? a/b for some a,b?Z, b?0.
• Therefore 3?3 is irrational.

19
Some Other Proof Strategies
Rosen 3.1
20
Backward Reasoning
We have used mostly forward reasoning strategies
up to now. However, sometimes it is unclear how
to proceed from the initial assumptions. Bacward
reasoning may help-- Motto If you cant prove
the original proposition, equate it to one you
can prove.
21
Prove ?(a,b ? Z, a?b)(ab)/2gt ?ab
(i.e., the arithmetic mean is always greater than
the geometric mean for this universe of
discourse.)
Backward reasoning proof (ab)/2gt ?ab Original
Assumption (ab)2/4gt ab Why? (ab)2gt
4ab Why? a22abb2 gt 4ab Why? a2-2abb2 gt
0 Why? (a-b)2gt 0 Why?
22
Prove ?(a,b ? Z, a?b)(ab)/2gt ?ab
Backward reasoning proof (cont.) But, ?a,b ? Z,
(a-b)2gt 0 implies a?b. We can now easily start
from a?b and go backwards to reconstruct the path
to prove the original proposition.
23
Conjecture and Proof
A conjecture is a plausible statement that has
not been proved. A conjecture may result from
recognizing that there are multiple examples for
which it is true. For some conjectures,
counterexamples are eventually found. However,
even if a conjecture is valid for very many
examples, this does not usually constitute a
valid proof. (Why?)
24
Conjecture and Proof
Sometimes a complex proof is constructed as a
series of conjectures that are then proved.
Sometimes a proof is found by referring to the
proof of a similar problem or class of
problem. There are many famous conjectures that
are still not proved (or only recently proved).
25
Fermats Last Theorem
xn yn zn has no solution for x,y,z,n ? Z,
x,y,z?0, ngt2 This is a conjecture made by Pierre
de Fermat (1601-1665), the French mathematician.
He wrote in the margin of his copy of the works
of Diophantus, an ancient Greek mathematician,
that he had a wondrous proof, but that it
wouldnt fit in the margin. He then died, leaving
no record of the proof!
26
Fermats Last Theorem
Attempts at proof over the years led to new
fields, such as algebraic number theory. Finally,
in 1994, Andrew Wiles provided a correct proof
that required hundred of pages of advanced
mathematics.