CHAPTER 21
The newest of methods of solving and calculating
Appendixes (I)
Contents
Solving congruence and Diophantine equations by "H.M" table (New Method):
Solving Diophantine equation of order in by "H.M" table:
A new and fast method for calculating
""
determinant (H.M method)
Definition of regular and irregular prime numbers by "H.M" determinant
New method of calculation of sum of
"k-th" power of the first "n" natural numbers by
"H.M" determinant (expressing "" via a determinant)
Determining the number of roots of perfect cubic degree of equation directly by "H.M" method
Proof of a new and applied "H.M" theorem (concerning the factorization of composite numbers)
(1)
We call number "n" as module, "x" as unknown, "a" as coefficient of unknown and "c" as constant number of congruence.
We know that if "" and according to
multiplicative inverse of "a" namely
(multiplicative
inverse of "a" in congruence):
Equation (1) can be written in below form:
Therefore, for solving equation (1), it is enough to calculate inverse of "a". We know that inverse of "a", (multiplicative inverse of "a") is calculated from below congruence equation:
(2)
In equation (2) if and
is dividing residue of
, we can
write:
(3)
Congruence equation (3) can be written in below Diophantine equation form:
(4)
According to "" and if
is dividing residue of
, we
can write:
(5)
As previous stage, congruence equation (5) can be written in below form:
(6)
As the same method, If we continue this action, coefficient of unknown will be "1" and with return to previous stages, we can calculate "x" with a reversible method.
We calculate all of above calculations with "4- row" table. This table has "4" rows that is completed after "4" stages.
For example we solve below congruence equation:
By using "H.M" table:
|
1378 |
621 |
136 |
77 |
59 |
18 |
|
|
2 |
1 |
|
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
|
|
621 |
136 |
77 |
59 |
18 |
5 |
3 |
2 |
1 |
|
|
537 |
242 |
53 |
30 |
23 |
7 |
2 |
1 |
1 |
|
Every table column, determines a congruence equation.
In this example, since, so we must calculate residue of
dividing
and
write it in second mesh of first row (from the left side) beside the module
.
Before explaining the table, we remind this point that for
solving every congruence equation, at first it is necessary to examine the
condition.
In congruence equation
, if
, so equation has answer when
"c" is divisible by
.
Therefore, for examining the condition, at first always we
calculate successive division in below form and write the obtained residue in
first row of the table respectively. If the first row of the table is ended to
number "1", it shows that condition
is established and if it is ended
to zero and number before zero is greater than "1", that number is
the greatest common divisor of "a" and "n".
Stage (1). We calculate successive divisions in below form and write obtained residues in first row of the table.
1999 |
1378 |
|
1378 |
621 |
|
1378 |
1 |
|
1242 |
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
621 |
136 |
|
3 |
2 |
|
544 |
4 |
, … , |
2 |
1 |
|
|
|
|
|
|
|
Stage (2). We write congruency’s constant number that
is "1" and "–1" successively in the second row of the table
(row)
from left to right.
Stage (3). We write the first residue (621) to the
last residue (1) another time in the third row (row) like the first row (ni).
Stage (4). Always we suppose the last mesh of forth
row in right side of table (row) is equal to constant number (+1
or –1) of its column in second row. It is necessary to explain that in first
row, with condition
, the number "1"
appears and so, division is finished.
For completing the forth row (row) of the right side in
successive "M" figures (according to figure and in direction
of arrows)
We act in this method (action of
every row, is written at beginning of every row):
We multiply the last number from the right side in forth row (537) in congruence constant number:
In the other hand, obtained residue of dividing number "537(1357)" or "728709" by "1378" is answer of congruence equation:
(Main answer of equation)
It is obvious that general answer of equation for arbitrary
integer value of is:
(General answer of equation)
Solve below congruence equation:
Solution. Since "", at first we write
obtained residue from division
in the right side of "module
19". Then we calculate successive divisions and write obtained residues on
first row of the table:
According to the above congruence, positive main answer of equation is:
;
|
19 |
7 |
5 |
2 |
1 |
|
1 |
-1 |
1 |
-1 |
|
|
7 |
5 |
2 |
1 |
|
|
|
-3 |
-2 |
-1 |
|
Find inverse of number "1999" with "module 1980".
Solution. Mathematics expression of problem is below congruence equation:
Since, we write residue of division
in the right
side of "module 1980" and then we calculate successive divisions.
Since always our purpose is positive answer of equation, therefore, it is enough to add number of module (1980) to number (-521) that the main answer of equation is obtained:
|
1980 |
19 |
4 |
3 |
1 |
|
1 |
-1 |
1 |
-1 |
|
|
19 |
4 |
3 |
1 |
|
|
|
-5 |
-2 |
-1 |
|
So, multiplicative inverse of number "1999" in "module 1980" is "1459".
Verify below congruence equation about existing or non- existing
answer:
Solution. At first, we obtain the greatest common divisor of two numbers "3953" and "2680" with successive divisors and writing the residues.
|
3953 |
2680 |
1273 |
134 |
|
0 |
|
1 |
-1 |
1 |
-1 |
1 |
|
According to the table:
Since constant number of equation (176) isn’t divisible by "67", so equation hasn’t answer.
Solve below congruence equation:
Solution. We solve the equation by forming the table:
()
|
77 |
69 |
8 |
5 |
3 |
2 |
1 |
|
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
69 |
8 |
5 |
3 |
2 |
1 |
|
|
-29 |
-26 |
-3 |
-2 |
-1 |
-1 |
|
One answer of equation from infinite answers is:
Since we want positive answer of equation, therefore, we add positive coefficient of module to this answer:
x=28
;
Find inverse (multiplicative inverse) of number "34" with module "41".
Solution. This problem is equivalent with below congruence:
We solve the equation by forming table:
|
41 |
34 |
7 |
6 |
1 |
|
1 |
-1 |
1 |
-1 |
|
|
34 |
7 |
6 |
1 |
|
|
|
-5 |
-1 |
-1 |
|
According
to the table, answer "
" will be obtained that for
finding positive answer; it is enough to add the module to it:
x
= 35
;
Therefore, inverse of number "34" is number "35" with module "41".
By using the table "H.M", solve below congruence equations.
1) |
, |
2) |
3) |
, |
4)
|
5) |
, |
6) |
7) |
, |
8) |
This table has a lot of usage’s that we explain the usage of table in solving linear Diophantine equations briefly. This table can be used for identifying the prime numbers and finding equivalent of exponential numbers etc.
We consider two-unknown Diophantine equation in below general form:
(1)
We write equation (1) in below congruence equation form:
(2)
Now, we solve equation (2) by table and determine value of. Then we
obtain value of
from relation
.
Solve below Diophantine equation.
(1)
Solution. At first, we write the equation in below congruence equation form:
(2)
Now, we solve congruence equation (2) by the table:
|
17 |
8 |
1 |
|
1 |
-1 |
|
|
8 |
1 |
|
|
|
-1 |
|
If we have a special answer of equation (1), we can write general answer of
equation, because if is a special answer of equation,
then:
If, by subtracting the two
equations of system:
In order to, it is necessary to have:
Therefore, if, answers of equation (1) will be
obtained from below equations:
According to equations (3), general answer of equation is:
Solve below Diophantine equation.
Solution. At first, we write equation in below congruence equation form:
(1)
Now, we must obtain product of (1340, 3953):
|
3953 |
1340 |
1273 |
|
0 |
|
1 |
-1 |
1 |
-1 |
|
Since the first row is ended to "Zero", it is results that numbers "1340" and "3953" have common factor "67":
In the
other hand since
, therefore we can simplify
equation (1) to 67:
Here, for solving equation (2), we form the table:
|
59 |
20 |
19 |
1 |
|
1 |
-1 |
1 |
|
|
20 |
19 |
1 |
|
|
|
1 |
1 |
|
According
to the table:
If we have value of, we can calculate value of
:
Therefore, the general answer of equation is:
Obtain positive and integer answers of below Diophantine equation:
(1)
Solution.
The greatest unknown coefficient is coefficient of
, therefore:
So, we can write:
(2)
With forming the table:
|
11 |
7 |
4 |
3 |
1 |
|
1 |
-1 |
1 |
-1 |
|
|
7 |
4 |
3 |
1 |
|
|
|
-2 |
-1 |
-1 |
|
According to below congruence:
Values of and
are:
As the same method:
(3)
In this congruence equation, coefficient of unknown is
"7" and the module is
according to congruence (2).
Therefore, answer of equation (3) according to the recent table is:
Equation (1) hasn’t integer answer for and
, so equation has two
kinds of special answers
and
that we can obtain
general answers of equation by using these special answers.
1) If, find integer numbers
and
so that below
equation is established.
2)
3)
4)
5)
We consider below "n-th" degree Diophantine
equation, we determine positive values of and
for it:
(1)
For solving equation (1), we write it in below congruence equation form:
(2)
It is obvious that condition of existing answer for equation
(2) or (1) is that, so, equation (2) or (1) has
"m" two by two non-congruent answers (different answers) with
module
.
In the other hand, if, we write equation (1) in below
form:
(3)
By using the table (H.M), we solve equations (2) or (3).
By using the table (H.M), answers or
will be obtained for
(2) and (3) respectively therefore, if
, we gain these answers:
(I) ,
(II)
Here, If value of exists, it is resulted from
relation (II):
(4)
If we consider and
limits of
will be determined:
(5)
Positive and integer values of
are determined according to (5)
and relation (II).
For example, we solve below fifth degree equation and we find its positive answers:
(1)
At first, we write equation (1) in below congruence equation form:
(2)
We simplify equation (2) and then solve it by table (H.M):
(3)
According to the table (H.M):
|
29 |
4 |
1 |
|
1 |
-1 |
|
|
4 |
1 |
|
|
|
-1 |
|
Therefore, answers of equation (3) are:
(4)
By substituting the values of in (1):
(5)
According to "" and "
",
Suitable value of
is determined from equation (5):
Here, by substituting in equation (4),
will be
obtained:
For another example, we solve below equation and obtain its positive answers:
Since we want positive answers of equation, therefore, we
determine limits of and
.
If "" ; then
or
that it is
enough to test the
main equation only for five primitive natural numbers:
Therefore, an answer of equation is "" and
"
".
Substituting another four natural numbers instead of (in main
equation), no other positive answer will be obtained.
If "" and "
", then
equation (1) is solvable easily, because, we can calculate limits of
and
in this case,
it means that we write equation (1) in "
" or "
" form
("
"and
"y" are positive):
We solve below equation:
Since we want positive answers of above equation, we can
calculate limits of and
from the main
equation:
By substituting three numbers that have been obtained for, value of
and then
will be
obtained. Therefore, we have determined the only positive answer of equation by
limits check method. In this method, it is enough to determine the limits of
:
If be a prime number, equation (1)
changes to below easy and solvable form:
(6)
We write equation (6) by using the Fermat’s (smaller)
theorem in
below form:
(7)
Answer of equation (7) can be calculated by using the table (H.M) easily.
We solve below 7-th degree Diophantine equation:
(1)
Solution. According to, we divide equation (1) by
"2":
(2)
We write equation (2) in below congruence equation form:
(3)
According to Fermat’s (smaller) theorem equation (3) changes to
below equation:
Therefore, answers of equation (1) are in below form:
Here, it is enough to calculate value of with respect
to
:
Answers of equation:
k |
0 |
1 |
2 |
… |
x |
2 |
9 |
16 |
… |
y |
5 |
683263 |
651914373 |
… |
For showing new method of calculation of determinant, it is enough to calculate some determinant in general and numeral cases. Therefore, we calculate third and forth order determinant in general and numeral case.
Calculation method. According to the determinant
rules always we can suppose that medium element of first column namely is non-zero
. Therefore,
we reduce
with
chain second order determinant method.
Action method. At first, we write second row another
time. In next stage, we put the obtained first column between second and third
column another time. Then we separate the obtained element to
"four-classes" in a determinant that we have a
determinant
so that one of its elements is a
determinant itself. At last, for
calculation of
, it is enough to divide the
obtained value by
.
Action stages. We write the second row another time and we put obtained first column between second and third column another time. Then separate the obtained element to "four-classes":
Stage 1:
Stage 2:
Every one of
"four-classes" forms a determinant. Then by calculation
of every one of
determinants and general
determinant and dividing obtained number by
, value of
will be calculated.
Stage 3:
Therefore is:
After necessary abridgment:
Calculate below determinant:
Calculation with second order chain determinant method
1) We write the second row another time:
2) We write obtained first column another time:
3) We have separated to second order determinants:
4) We calculate the second order chain determinant:
5) We divide the second order chain determinant product by middle element of determinant’s first column:
6) Calculated value of:
Here, we present simplified and brief method of second order chain determinant. At first, we separate first column and perform actions on successive rows
in below form:
(According to the determinants rule, we suppose that "")
Here, we calculate example (1) (chain determinant of second order) by simplified method.
At first we separate the first column and perform actions on
rows and then we calculate "".
2) Calculation of forth order determinant in separation method (chain second order determinant):
With the same method, at first we reduce to
and then reduce
to chain
second order determinant like before.
For separation "" to second order
determinants, it is enough to write second and third rows another time and
write obtained first column alternately repeatedly and then we perform the
separation. Here, according to determinants rules, we suppose
,
(like with
changing two rows or two columns). Therefore, always we can suppose that "
,
", then
with this hypothesis:
According to the determinant rules, we can suppose that
""
and so:
Now, we present simplified method of chain second order determinant:
At first we separate first column and then we perform the
actions on rows in below form (we suppose):
(Final result)
It is necessary to explain that for calculation of the
"n-th" order determinant when, simplified method of chain
second order determinant can be used also, because for higher order of
determinant, we need a method with higher speed. For calculation
, we must
attend that first column middle elements of every stage, namely
,
,
and … must be
non-zero. For example for calculation
, we must have
; because in
stage(1) of
,
we must have
, and in stage (2) of
, we must have
and
in stage (3) of
, we must have
. It is clear
that according to determinants rules for every stage, this thesis is certain
and it hasn’t any disorder to generality of problem.
Calculate "forth order" determinant by chain second order determinant method.
Calculation with chain second order determinant method:
Here, we calculate example (21.4.5) by simplified method of chain second order determinant.
In order to this, at first, we separate the first column and perform the calculation on rows in below form (like third order determinant):
Calculate below "fifth order" determinant.
Solution. We perform calculation on rows and then we
obtain,
and
determinant.
Product of determinant is:
Below Congruence is result of theories related to verification of Fermat’s last
theorem:
Here, Congruence is equivalent with congruence
and below theorems are resulted
from congruence (1).
Number is irregular if and only if a number
exists for
so
that
is
divisible by
.
Number is regular if and only if for
every
,
number
isn’t divisible by.
It is easier to consider the general sum:
It can be proved easily with mathematic induction that
numbers,
values of polynomial
of
degree for
that has rational
coefficients (
: Bernolli’s numbers):
It is interesting that Bernolli’s numbers entered into mathematics during solving one of astronomy problems.
The subject is that infinite sequence of Bernolli’s numbers appears during converting some simple functions as power series. Like, we can prove:
This series was put against Bernolli during astronomy examinations.
This point that in series related to extension of , only
Bernolli’s numbers appear that have an even index, isn’t randomly, because we
can prove easily that all of Bernolli’s numbers that have odd index (except
number
),
are equal to "zero":
If then
Bernolli’s numbers with even index, have interesting
characteristics. Like values of famous "Riemann's function" (function
), can be
expressed with respect to Bernolli’s numbers when have even power:
The first Bernolli’s numbers are in this form.
The numerators of these fraction increase rapidly, For example:
Suppose:
is indicator of number. Without special difficulty we
can prove that for
, denominator
from number
isn’t
divisible by
.
But, for every, we have this equality:
It means:
Therefore, for, the number in right side of
parenthesis is an integer number.
We return to previous congruence (module) and we simplify it to
, so, the
result is:
Therefore, congruence "" is established if and only
if we have:
Therefore, we can present Kummer's theorem.
Prime number is regular if and only if it is
not a divisor of numerators of Bernolli’s numbers in "
" form.
We express some examples (see the Bernolli’s numbers that we expressed before):
isn’t divisible by "5".
and
aren’t divisible by
"7".
and
aren’t divisible by
"11".
and
aren’t divisible by
"13".
and
aren’t divisible by
"17".
and
aren’t divisible by
"19".
and
aren’t divisible by
"23".
Therefore, all of these powers are regular.
In other word, Fermat’s last theorem is correct for powers "5,7,11,13,17,19 and 23".
In next section, you will be acquainted with "H.M"
new method of calculation of that is expressed by a
determinant.
We present calculation by determinant without proof ("H.M" method):
Since is a prime number and
, so
.
Number is a regular prim number if and
only if
is
not divisible by
for every
.
In fact, according to identification function of prime numbers, we can write:
Therefore, set of regular prime numbers is defined in below form:
(Set of regular prime numbers)
It is obvious that according to definition of set of regular prime numbers, set of irregular prime numbers can be defined by definition of prime numbers set:
(: Set of irregular prime numbers)
Here, sets of regular and irregular prime numbers are
defined by expressing "" and definition of prime
numbers set "IP".
Calculation of "". New method for
calculating the sum of "k-th" power of
natural numbers
by
determinant
Here, we prove that "" calculates by below "k-th"
order determinant:
Fermat’s last theorem is correct for powers.
Other sets can be defined by prime numbers formula.
Here, sets of regular and irregular prime numbers are
defined by expressing and definition of prime numbers
set "
".
Here, we prove that "" can be calculated by below
"k-th" order determinant:
Proof: We know that if then:
We put,
and
in above identity:
Now, if we put values respectively in above identity,
we have below equality with adding obtained relations:
We write above equality in below form:
Now, if we put values respectively for
, below system
will be obtained
:
So:
(1)
In equation (1), if we have:
then equation has a real root that is computable easily,
therefore for calculating the root of equation (1) it is enough to perform this
conversion and
establish the above mentioned conditions. As we know, after conversion we will
have below equation:
(2)
Now, for calculating, we establish the first
condition namely
. After establishing this
condition we will see that value of
is not obtained.
Now, we establish the second condition, namely. And we will
see that value of
will be calculated. Therefore:
(3)
After summarizing relation (3), we have below second degree
equation (with respect to):
(4)
Here, for calculating the value of, it is enough to solve
the second degree equation (4). After solving equation (4), the value of
is:
(5)
Discriminant of cubic degree equation (1) is
obtained from value of
:
(6)
Here, with obtaining such equation, we can discuss about the number of its roots:
1) If, then equation (1) has a real
root and two complex roots.
2) If, then equation (1) has an
additive root and one simple root.
3) If, then equation (1) has three
real roots.
In equation (1), If, then the equation is:
(7)
It means the focal equation of cubic degree will obtained.
Now, if we put value in Discriminant
of equation
(1), Discriminant
of focal equation (7) will
obtained:
If is divisible by "p"
and result of
is an odd number, then:
Proof. For every and
, we can write:
(1)
If is odd and according to below
equality for every odd
:
(: Number integer part)
( is odd number)
So:
(2)
If is prime number, according to Wilson’s
theorem, expression
is divisible by
and
conversely (If
is divisible by
and
is a prime
number). Therefore, if
is factorization in relation (2)
form,
is
a prime number:
(3)
I. If then:
(p: prime number)
II. If then: