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Introducing prime integers

The prime integers are special with respect to
multiplication.

2 is a prime integer.
3 is a prime integer.
4 is not a prime integer.

Why is 4 not a prime integer?
It is because 4 is 2 * 2.

5 is a prime integer.
6 is not a prime integer.

6 = 2 * 3.

4 and 6 are called composite integers because they are the
product of smaller magnitude integers.

The integer 1 is very special. It is named "unit" because it
divides every positive integer. To divide by 1 is the same
as not dividing.

For this reason, we call 1 the multiplicative identity.

Multiplying by 1 and dividing by 1 give same result.

A prime positive integer has two special properties, from
which all of its other properties are derived.

(1) A prime is not a product of smaller magnitude integers.
Thus if a number is composite, it is not prime.

(2) If a prime divides the product of two positive integers,
then it must divide at least one of them.

Within the integers,either of these two properties could be
taken as the definition of prime integer.

Numbers with property (1) are called irreducible. A positive
integer has property (2) if and only if it is irreducible.

In most other number systems, this is not the case.

In most other number systems, property (2) is the preferred
definition for prime numbers. In those number systems,
numbers with property 1 are called irreducible.

In most number systems primes, numbers with property (2), are
aways irreducible, meaning have property (1).

In those number systems, prime numbers are always
irreducible, but irreducible numbers are not always prime.

The negative integers are "associates" of the positive
integers.

An associate of a number is that number multiplied by a unit.

-1 is also a unit. It is a unit because 1/(-1) is an
integer, namely -1.

-1 is an associate of 1.
-2 is an associate of 2.
-3 is an associate of 3.
etc

The integers are partitioned into three parts: units,
composite integers, and primes.

We define the primes by what they are not. Primes are
neither units nor composites. In a time past, primes were
defined as not composite. Only after experience with many
other arithmetical systems did mathematicians realize the
importance of distinguishing units from primes.

Today, math textbooks carefully define positive prime
integers as integers greater than 1, not expressible as a
product of smaller integers.

Why has the study of the prime integers become as important
as they have? It is because we can explain much of the
nature of the integers in terms of the prime integers.

To truely understand the numbers 1,2,3,.....
we need to understand many things about prime numbers.

Remainder arithmetic language has become an essential tool
for working with prime numbers.

0 * 0 = 0 mod 5
1 * 1 = 1 mod 5
2 * 2 = 4 mod 5
3 * 3 = 4 mod 5 because 3 * 3 = 9 = 4 + 5
4 * 4 = 1 mod 5 because 4 * 4 = 16 = 1 + 3 * 5

A consequence of this is that no square integer can ever have
remainder of 2 or 3 when divided by 5.

Similarily, no square integer can ever have a remainder of 2
when divided by 3.

No square integer can ever have a remainder of 3,5, or 6 when
divided by 7.

No square integer can ever have a remainder of 2 or 3 when
divided by 4.

No square integer can ever have a remainder of 2,3,5,6, or 8,
when divided by 9.

When we divide a number by 10, we get, as remainder, its
rightmost digit.

What rightmost digits are never the rightmost digits of
square integer?

We can answer this question by looking at the squares of the
10/2 = 5 integers, starting with 0.

0**2 = 0. It is possible for a square to have right most
digit of 0.

1**2 = 1. It is possible for a square to have right most
digit of 1.

2**2 = 4. It is possible for a square to have right most
digit of 4.

3**2 = 9. It is possible for a square to have right most
digit of 9.

4**2 = 16. It is possible for a square to have right most
digit of 6.

5**2 = 25. It is possible for a square to have right most
digit of 5.

6**2 = (10 - 4)**2 has the same last digit as 4**2.
7**2 = (10 - 3)**2 has the same last digit as 3**2.
8**2 = (10 - 2)**2 has the same last digit as 2**2.
9**2 = (10 - 1)**2 has the same last digit as 1**2.

The only posible last digits, in base 10, of squares are
0,1,4,5,6,9. A positive integer, in base 10, that has right
most digit of 2,3,7, or 8, is not a square integer.

A very important question for the analysis of arithmetic is,
if p and q are two primes, is p the remainder of a square
when divided by q, and is q the remainder of a square when
divided by p?

When p is the remainder of a square when divided by q, we
call p a square residue mod q. Our question becomes:

If p and q are two primes, is p a square residue of q, and is
q a square residue of p?

This question can be answered by knowing:

(1) If one of the primes has remainder 1 when divided by 4,

then p is a square residue of q if and only if q is a square
residue of p.

(2) If both of the primes have remainder 3 when divide by 4,

then p is a square residue of q if and only if q is not a
square residue of p.

(3) 2 is a square residue mod prime p, if and only if

p = 1 mod 8 or p = 7 mod 8.