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Well-Ordering Axiom

Well-Ordering Axiom: Any nonempty set of positive integers contains a least element.

The minimum is another term for least element. The largest element is the maximum. An important note to remember is that the integers do have an order (but no minimum or maximum)! Also, the Well-Ordering Axiom is at the center of some controversy. It is equivalent to the Axiom of Choice and thus the root of the Continuum Hypothesis. See numbers lesson 13 for more details.

Prime Factorization

Once a natural number has been factored into prime factors, we can write its prime factorization (also known as prime decomposition). When we do this, we list each prime factor in increasing order and indicate how many times it is repeated by using a superscript as an exponent. For example: 60 = 22•31•51. When done this way, the prime factorization for the natural numbers is unique. The associated prime factorization theorem (or Fundamental Theorem of Arithmetic) could be proved, but not here. We can use prime factorization to find Greatest Common Factors and Least Common Multiples. Another method is Euclid's Algorithm (a procedure) which we intend to link to here eventually.

GCF: Greatest Common Factor (or GCD) is the greatest number that divides two given numbers.

For example: The factors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} and the factors of 12 are {1, 2, 3, 4, 6, 12} and so the factors 30 and 12 have in common are {1, 2, 3, 6}. The GCF would then be 6.

Two numbers are relatively prime if they have no common factors (excluding 1).

In other words, two numbers are relatively prime if their GCF is 1. Examples are: 15 and 16, 20 and 21.

LCM: Least Common Multiple is the smallest (positive) number which is a multiple of two numbers.

The definitions of GCF and LCM could be extended to more than two numbers. In fact, since the calculator will only do pairs, such an extension gives more meaningful test questions!

Example 1:   The multiples of 4 are: {4, 8, 12, 16,...} and 6 has multiples of {6, 12, 18, 24, 30, ...}. The intersection of these sets is {12, 24, 36...}, so the LCM is 12.

Example 2 (Using Prime Factorization):   30 = 21•31•51 and 12 = 22•31. Thus the GCF(12, 30) is 21•31 = 6 and the LCM(12, 30) is 22•31•51 = 60. Notice how we choose the smallest exponent for each prime factor. It might help to note that 12 = 22•31•50 and remember that anything to the zero power is 1. Note how GCF(12, 30)•LCM(12, 30) = 6•60 = 12•30!

Example 3:   25 = 52•170 and 85 = 51•171. The GCF(25, 85) is 51•170 = 5 (choosing the smallest exponents) and the LCM(25, 85) is 52•171 = 425 (choosing the largest exponents).

Primes Form an Infinite Set

It can easily be shown that the set of prime numbers is infinite. This proof, which dates back to Euclid, is as follows. Suppose, on the contrary, that there are only finitely many primes denoted p1, p2,...pn. Form the product N = p1p2p3•...•,pn. Then, the number N+1 is not divisible by any pi and so must be divisible by a prime other than these (including possibly N+1 itself). This contradicts our original hypothesis that we listed all the (finite set of) primes, hence this hypothesis is false. Hence there must be infinitely many primes. This is a classic proof by contradiction. It remains an open question whether or not there are an infinite number of twin primes. Using the well-ordering axiom, we can also prove all numbers are interesting!

Sieve of Erastosthenes

Having established the fact that there are infinitely many primes, we might want to generate a list of primes, or determine if a given number is prime. Erastosthenes, a Greek mathematician around 200 B.C., created a simple algorithm to find primes. The procedure represents a sieve, or device used for sifting out grains, since he actually punched holes. The method is simple:
  1. Write down the numbers from 1 to 100 (or any desired range).
  2. Start with two (the first prime number).
  3. Eliminate all its multiples.
  4. Move to the next prime (the next number on the list which you have not eliminated).
  5. Go back to step 3 and repeat as many times as necessary.
Note that anything above [square root of 100]=10, does not eliminate any more numbers, since factors come in pairs of a big and a small.

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Division Rules

Here are some useful rules for quickly checking for divisibility of numbers by small factors.

Divisibility by 2: If an integer is even, that is ends in 0, 2, 4, 6, or 8, it is divisible by 2.
Divisibility by 3: If the sum of the digits of an integer is divisible by 3,
then the integer is divisible by 3.
Example: 729 => 7 + 2 +9 = 18 => 9. Thus 729 is divisible by 3.   Note how this was done recursively.

Divisible by 4: If the last two digits of the integer are divisible by 4, then the integer is divisible by 4.
In general, an integer is divisible by 2n if the last n digits are divisible by 2n.
Divisibility by 5: If the last digit is 0 or 5, the integer is divisible by 5.
If the last n digits are divisible by 5n, then the integer is divisible by 5n.
Divisibility by 9: If the sum of the digits of an integer is divisible by 9,
then the number is divisible by 9.

A common method taught in days past for finding computational mistakes was called Casting Out 9. This is really a form of modulo arithmetic. In other bases, this method extends to "Casting Out base - 1".

Divisibility by 11: If the sum of the digits in the even powers of 10 positions differ from the sum of the digits in the odd powers of 10 positions by a multiple of 11, the integer is divisible by 11.

Example: 1,234,508 => 1+3+5+8=17 and 2+4+0=6, thus since 17-6=11, 1,234,508 is divisible by 11.

In general, determining if a large number is prime or composite is a difficult task. Substantial research continues in this field due to the fact that many encryption schemes are dependent on this difficulty.

Perfect Numbers and Mersenne Primes

A perfect number is equal to the sum of its factors, excluding itself.

The first two perfect numbers are:
6 = 1+2+3 = 1•6 = 2•3 = 22-1•(22 -1) and
28= 1+2+4+7+14 = 1•28 = 2•14 = 4•7 = 23-1•(23 -1).

The ancients considered these perfect partly due to their close proximity to the number of days in a week (which is not celestial!) and the lunar/menstral cycle.

Mersenne Numbers are of the form 2n - 1.   Mersenne Primes are primes of the form 2n - 1.

Marin Mersenne was a 17th century monk who studied the numbers 2n - 1. These can only be prime if n is prime, but that is no guarantee of primality as seen in the homework. Euclid showed that the known perfect numbers were of the form 2p-1•(2p-1). Euler proved that even perfect numbers could only be in this form. It remains an open question as to whether there are any odd perfect numbers. Whenever another mersenne prime is found, another perfect number is generated. The largest of the 40 now known has n=24036583. It was just found last May (2004). The largest known prime is usually a mersenne prime. The GIMPS project is a way the author and some students are involved in this search.

Cardinal vs. Ordinal Numbers

Cardinal Numbers are positive integers (counting numbers) that represent "how many?"
Ordinal Numbers are numbers that describe position: first, second, third, fourth,... last