# The Laws of Prime Numbers and Odd Composite Numbers

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The Laws of Prime Numbers and Odd Composite Numbers
Aixinjiaoluo . Xi Guo-wei
Fed . 2004
Reg No : O7-2005-A-271
By The Copyright Bureau of Jilin Province , CN
The Preface
About 330～275 B . C Euclidean put the mathematic law of expressing prime numbers , and for 2200 years so far there is no results for them.
The law of holography of numbers of《Theory on Motion》tells us that the infinity in Math. only is the relative infinity , not absolute infinity , and it is impossible to seek for the numbers of absolute infinity , and it’s wrong idea. The concept of infinity is the concept of motion in essence , of course , the numbers are produced in motion . The first ones are irrational numbers , and the integers only are tiny spots among them.
Also《The Theory on Motion》found in Lucas Number the formula of “the new formula（M）of remainders”, and from this derived the following :
M = a r/r              ——（M）
a =（5 1/2+1）/ 2   “r”is an odd number and leading to the laws of prime number and odd composite numbers.
The Discriminant of Prime Numbers and Odd Composite Numbers
A． Division and screen analysis :
1. The dividend“b”is divided by“a”and gotten the quotient“q”, and the relation of which can be expressed in form of fraction , as follows :
——（q）, and when a、b、q are all positive integers the“b” can be exactly divided by “a” , and this relation of division can be thought in an images as“screen”.  The“b”can be divided by“a”, in an analog the“a”be screened and gotten“q”.
2. When“b”can’t be exactly divided by“a”there is a relative formula :
b = a ? q + c      ——（b）
1≤ c ＜ a , c is the positive integer in one hand ,“b”can’t be exactly divided by“a”, or in others ,“b”has no integer’s factors , and an images drawn that“b”can’t be“screened”by“a”.
3. The binomial expansive co-efficient formula is called“binomial coefficient screen”:
“q”is the positive integer.
The denominators are the factorial numbers increased by“k”, and the numerators are the continued multipliers by successive decreasing“k”, and“n”is the binomial involution number (exponent).
The“k”is the number of term of the binomial expansion and
4. The factors of minimum prime numbers of the positive integer “N” is not greater than  .
The“N”divided by the positive integer , less or equal to  , would confirm soon that whether“N”has factors of the integers or not. The learner Eratosthenes earlier set up the “screen”of the prime numbers by this kind of division.
B. The Definitions of Prime Numbers and Composite numbers
1. One positive integer , which can be exactly divided by“1”, and itself , is called as a prime number . or one positive integer , which only has“1”and two factors of its itself , is called as a prime number.
2. One positive integer , not only is divided by“1”and itself , and other positive integers , is called as a composite number , or one positive integer , which at least has 4 or more than 4 factors of numbers , is called as a composite number.
3. All the even numbers (except 2) are composite numbers , and in fact , the prime numbers only indicate prime numbers in the odd numbers and are called as odd prime numbers , and the composite numbers in the odd numbers are called the odd composite numbers , and
4. See from the above that there are only two kinds of numbers in odd numbers : the prime numbers and the odd composite numbers.
C. The Distributive Law of Multiplication
Here is formula :
m?(a + b + c) = ma+mb+mc or ma + mb + mc = m?(a + b + c)
There are the same factors (m) in the terms of algebraic sum , first all , take the different factors in all terms for algebraic sum operation , then take the sum numbers multiplied by the same factors (m) for multiplication operation , and the results are same , and vice versa.
D. The Binomial Expansion and its Coefficient and the Special Formula :
1. The expansive formula of binomial theorem.
(a+b) n= ( ) an?bo + ( ) an-1?b1 + ( ) an-2?b2 + … + ( ) an-k?bk + … + ( ) an-n?bn =  an-k?bk             ————
2. The binomial expansive general term formula of coefficients
( ) =  —— ( ) o
(kM , for the “r” is the positive integer (odd number) , indicating “ar” can be exactly divided by “M”.
As clear as the above :
always has  r   ar  but here   , M/ar
(2) To substitute the equation (r)M into (rm) , and result gotten :  and is named “the share of integer   occupies the proportion of the whole number of ar ”.
(3) Observe the relation between the shares of the whole numbers and decimal shares . The equation (rm) divided by “ar”,  relative factor of the “r”, gets the integer shares :
The equation (r ) divided by “ar”, the relative factor of “r” , and gets the decimal shares :   which are certified that :
(a). The shares of the integer :   and the shares of the decimal :   can be completely separated at the time for that the share of integer is “1” (max.probability) minus the share of the decimal , which is not straight expressional numbers , and just the decimal share is the straight expansive numbers . From this surely confirmed that the integer’s part of M is the “m”, and the decimal part of “M” number is just “ ” number.
(b). Because the results of integer “m” and the decimal “ ”, that is to say , the integer part (rm) has been divided exactly by the odd number “ r ” and in the binomial expansive equations , seemly to say , the “ r ” number can’t be divided in all coefficients ( ) of the whole equation , then “ r ”must be the prime number , Or in another way , as “r” can be recomposed from [ Lr -1 ] formula , the number “ r ” is just a prime number , and “ m ” is a positive integer .
(4) The integer “m” is the max . relative factor of (Lr-1) , including all the factors except “r” ; and the decimal △ is the min . relative factor of (1+a-r) , for the comparision :
(Lr-1)﹥(Lr- )     1﹤ ﹤r
(1+a-r) ﹤( +a-r)
When  =1, it’s :
(Lr-1)=(Lr-1)max
(1+a-r)= (1+a-r)min
——
when (Lr- ) , ( +a-r) taken , 1<
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