The Characteristic Extension Of New "Remainder Formula (M)"

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The Characteristic Extension
Of  New “ Remainder Formula (M) ”
Xi Guo-wei
The formula    ——(M) is derived from Lucas Numbers , but different with Lucas Number itself .
In which  ,and in the formula the exponent “ ” of “ ” and the divisor “ ” in M are the same numbers , and can be expressed in two forms :
a. the exponential form       and
b. logarithmic form
The two kinds of “ ” formulas in a and b are all inverse functions of M , and so the positive function only has one (M) formula .
A . The first one of the inverse functions of (M) , the evaluation method of “ ” of exponent form in “ ” and is expressed in  “ ” :
for the specific implicit function the best method of evaluation is the limitless iterative method。
firstly , take the  =1 , the result is :
then , take “ ” again by “ ” , result is :
then take “ ” by “ ” again, the result is :
When   , the following formula holds water :
Attention to : this is “ ” value for the (M) value (certain) , and after the (M) changes   change into another fixed value , and the every “ ” is produced by man’s substitution of limitless times “n” , and this kind of iteration can be expressed in this :
In which each   is the power exponent of the next “ ” .
The above two expressions are very complicated , and a specific “signs of function relation ”given：
——
and   is from   to
in which the “ n ” is the iterative times by man , trending to infinity , and each  “ ” only corresponds to one value of “M” , and when   then   .
To take “ ” as “1” can make the frequent evaluation of “ ” , and shorten the midway for calculation for   。of course , also take “ ” as the other arbitrary positive numbers .
The Tablet 5 gives the   values of M from the smallest values “1.400794211” to “M=13” for reference :
(M≥  ,  ≤1)
The Tablet 5
M	
M	
1.400 794 211
a
2
3
4
5
6
7	1.400 794 211
1
0.700 397 105
0.405 073 81
0.287 028 805
0.222 614 63
0.181 914 185
0.153 833 63	8
9
9.242592624
10
11
12
13
0.133 299 64
0.117 579 082
0. 114 313 159
0. 105 192 285
0.095 169 219
0.086 891 648
0.0799 398 03
0.595 664 113
(only take the ninth decimals ) and the Tablet only gives the some “ ”values of specific M:
(1) When M1.4=1.400794211 , the “ ”…, and beginning from the point , if the “M” is still the positive number of greater than “1”, then use the logarithmic form b:
, and calculate the “ ” when M＜1 , and it is not necessary to use   the formula , of course , also continue using   formula , and calculate “ ” at the time of M＜1 . For the   the point found , indicating the exponent form “ ” and logarithm “b” are united , and they have the different forms for the different spheres of M.
(2) When   , the  and   is very valuable for the applications by Fibonacci Numbers Sequence , so the Tablet given , and
(3) When M=9.242 592 624…
Let
then     ( )
then
The formula here is only for prescription .
B . The second of the inverse function of (M) formula :
The “ ” evaluation method of “b” logarithmic form (“ ”is no “′”):
the limitless iterative method still used .
firstly take   , the M is some fixed value :
Take   , then
, and
Take   , then
When   , still known :
After the limitless times of iterations every M valuation corresponds to one value “ ” , to decrease overlaboring the specific signs of function relation given :
——
and the “ ” expresses from “ ” to “ ” , and the “n” is the iterative times by man , trending to infinity .
Each one “ ” only corresponds to one value “M” , and when   , then the “ ”≥1 , and
the Tablet 6. gives the   values ( ,  ≥1) from M=1.400 794 211… to M=13 .
The Tablet 6.:
M	
M	
1.400 794 211…
2
3
4
5
6	1.400 794 211
1
4.621 296 312
5.028 737 375
6.009 853 786
6.892 427 769
7.543 794 778
8.060 296 004	7
8
9
9.242 592 624
10
11
12
13	8.488 103 412
8.853 080 486
9.171 207 596
9.242 592 624
9.453 058 234
9.705 993 523
9.935 344 347
10.145 095 04
（Only take the ninth decimals）
From the Tablet 6 seen :
(1)  ≥1  (when M≥ )
(2) When M1.4=1.400794211 then
the point is just at the subangles 45°,exactly is:
[ ]=f[ ]
Indicating here the two curves ,f meet at the point  = .
The  [ ] is in the field of r value ≤ 1.400794211, and f[  ] is in the field of r value ≥ 1.400794211.
(3) when  =9.242592624… then   = 9.242592624…
=
And the point is at the subangle 45°.
Also seen:  ?  =   =
=
Here ,the two functions  [ ] and f[  ] are drawn into connecting curves at the coordinate roM,(see the Diagraph 17) according to the data in the Tablet 5 and Tablet 6.
This is one of very specific curves.
C. The Evaluation Method of Positive Function (M) and lts curves .
If put (M) formula :  in the same coordinate roM of   and   at the time seen :
The coordinate “ ” and the coordinate “M” have been exchanged Mor .
The (M) evaluation is normal , and the each “ ” value given corresponds to a value “M” , and make the curve M of o ≤ r ≤ +∞ , as seen as the Diagraph 17 , and the (M) curve and the   curve are exactly symmetrical about the line of   angle .
Here the (M) curve is made by the function relation , is not like that one curve from the two functions   , and   .
D. The Function (M) and the Nature of inverse Function   , and   and Their Relations .
(1) The function (M) has the smallest values :
,
Here “ ”is the derivative of “ ” ,
and the   of extreme value , So
（—  only is the   value of the  ）
The prescription to   ,and   :
(1) When  ＞  or  ＜
The M value in the (M) formula increases and when  then the  , and when   , then   .
(2) As for   and   and (M) , then the M＜  doesn’t exist , and when M＞  ,
Then :
This is very important nature .
So , here seen :
——( )
and for
So ,
——( )0
The result is   formula indicating that :
From the numerator and denominator in formula ( )0 plus or minus “ ” and   at the same time , their quotient doesn’t change , in which “n” and “m” are the arbitrary real numbers , imaginary numbers , and their sphere is very large .
Seen from formula (M) that the M value can be any positive number , that is “ ” value can be positive ( greater than or less than  ).
Seen from the Algebra when only researching the positive value :   and M
If
It is the smallest value of ratio   .
For (M) function is continuous and derivative , and the motion intervals of the ratio   is :
1 ≤   ≤ +∞
And can take any values in the intervals .
For ex: Take  =2 , seek for values :   and
at the meanwhile :
Then ,
So , seen from ( )0 :
The positive number ＞1 can be expressed as the division relation :
(1 ≤ N ≤ ∞)
By the algebraic law of division known that :
But , if written as :
When :
It is “abnormal” . So the formula ( )0 is so hard understandable and believable , but it’s the fact , and has “destroyed” the division law of algebra .
The key is that :
The “ ”and  “ ” are not the “m” an “n” expressed by division and can be in arbitrary numbers :
m≠  ,  n≠
Its specific values are equal .
For ex :
If   , extract a root :   ,   :
Here  ≠7 ,  ≠3
So
Then
And
So seen :
But              ——（M）
Obviously , a basic algebraic principal :
——(NW)
When N﹥0 , and let
Then     ——(W)
“The N times of the number W are equal to N-order powers of W .”
When N﹤0 the left of (NW) formula is substitute by “—N”
The right of (NW) is substituted by “—N”
So , the left of (NW) formula are equal to the right of （NW）
From the above known that :
Whether N﹥0 or N﹤0 the N times of the number W are equal to N-order  powers of number W .
When N﹥0, NW=Wn are the real numbers , and when N﹤0 , NW=WN are equal to the imaginary numbers .
When
——
The imaginary unit : i=  , from here known (NW) formula is the algebraic original reasons produced from the imaginary numbers , and the
connecting the Numbers Theory and complex numbers .
When
( infinitesimals)
=e (Napier’s Numbers )
=2.718 281 828 459 045…
When    ( =infinitesimals)
=e  (Napier’s Numbers)
From the above known that :
(NW) formula is the original root of limits produced and connecting the Number Theory and calculus .
In fact the base of the calculus the only “e” is a limit developing on the limit “e”, for another important limit :
In which the sine is a function of the limit “e” also :
( )
So if rather the calculus are developed on the 2 important limits than that the calculus only is the extension and derivation of the limit “e” .
When the formulas (NW) 、(W) and (M) and (m),   and “e” in then found and the algebra is the conclusion on  united original root about complex numbers , complex function , differential and integeral . It is quite reluctantly to name these “relations ” as “connect” and seen from the some formulas that ： the basic main elements of higher math . in the algebraic laws indicating its gigantic features , and they are not the parallel relation , but the relation of root , stems and branches .
(2) The three curves (M) , r=f(M) and   intersect on the two points:
1.400 794 211…=M
9. 242 592 624…
They are the specific relative points of the three functions , and here :
Also here :
And the result :
——（ ）
Or :
And from （ ）gotten :
——（ ）
The   and   of the arbitrary n values for calculation still have to use   ,   formulas :
If the   and   values of  n=3,4,5,6,7 are given in the Tablet 7.
The tablet 7:
3
4
5
6
7	    18.029 560 88…
27.569 711 06…
37.718 974 8…
48.361 777 65…
59.416 725 91…	1.215 221 43…
1.148 115 22…
1.113 073 15…
1.091 485 11…
1.076 835 41…
Also seek for more values of    and   of  .
A specific phenomenon seen from the formula ( ):
So ,   order powers of the   are equal to the   order powers of the   .
The “law of prime Numbers ” curve consisting of the curves (M) , f(M) and   is just the type of n=2 , and when   , then   ,   .
The attention to :
1.400 794 211… ≥ ≥1
That is to say , the limited fields (1≤ ≤1.400 794 211…) corresponds to limitless field ( ) , but they are semi-closed .
The curves of (M) , f(n) , and   cross the segment (1≤ ≤1.400 794 211…) have some ( ) , and every function curve of n value has two points intersecting on the line :
.
The so-called the curve of “ The Law of Prime Numbers ” is just for obvious , in fact the law of prime number is just on the partial curve of “ ” when n=2 , and also without minus   from it ,and is not obvious .
E. As for the formulas : （ ）and ( )
we can express the formula(m) of the law of prime numbers as the form  ( ) :
then :
Here are :
Obviously :
So
In which
——
Just the same :
The odd number   , which can make “m” be a positive integer , must be a prime number ( ≥3) , or it (the  ) is a composite number .
For ex : suppose that :
=7 , seek for the formula   substitution after “n” :
Here :
is the prime number .
The above gives the function of  “  number” .
From the hyperbolic function relation known that the real numbers are all expressed as formula :
(except the imaginary numbers )
The De . Moivre’s Theorem :
So ,
The “ ” order powers on the both sides :
For :
Also the result is :
That is the ( ) formula .
The ( ) formula certifies from the known hyperbolic function , just so . The single mathematic relation doesn’t deduce the result ,especially express no existence of the “ ” ,without the new mathematic meaning .
The (m) formula also can be expressed as the hyperbolic function , in which :
Then
F． The Two shapes of the Lucas Number Sequence :
If let   , the “ ” including “ ” as the odd numbers and “n” as the even numbers, and can be written :
For   ,
then:
So seen :
——（u）
The two same kinds of more extensive the Fibonacci Number Sequence closed and the “ ” is cancelled in the deduction , and known the Lucas Number Sequence is just a (u) formula on “ ” , so the (u) formula can be arbitrarily developed .
G . The simple relation between the Lucas Numbers and Fibonacci Numbers :
Except   the coefficient relations of integers have been found :
The Lucas Numbers   ,
When  n ≥ 2
The recursion relation seen :
,   ,
Attention to :
…… …… ……
That is :
——（ ）
Abstracted from PP. 400~417 of 《The theory on Motion》