This is Math Newsletter number 7; Wednesday, September 8,

2010. If you send any part of this newsletter to a friend,

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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.