yarosh gen fermat's comments
DESCRIPTION
paperTRANSCRIPT
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V.S.Yarosh
The state unitary enterprise All-russian research institute For optical and physical measurements (sue VNIIOFI)
RUSSIA,119361, Moscow,Ozernaya,46. [email protected]
Non-modular elliptical curves as established fact and
as result application Abel group and Diophantine equations for solutions
Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and
Swinnerton-Dyers. (Survay problems)
Abstract If whole numbers v and u are such that v > u and
greatest common divisor GCD (v, u)=1 , at that v and u of different evenness, than triads ooo c,b,a , generate a endless series of dyophantines
equations:
220
0
220
uvcuv2buva
+==
=
as endless series a primitive solutions for Pythagorean equations: 2o
2o
2o cba =+
and as endless series irrational roots:
n n222n2222n22222
n no
2no
2o
2no
2o
n 2n222n2n222
n 2no
2o
no
2n2o
n 2n222222n22n22
n 2no
2o
2no
2o
no
3/])uv()uv2()uv()uv()uv[(
3/)cbcac(c
3/])uv()uv2()uv2()uv()uv2[(
3/)cbbab(b
3/])uv()uv()uv2()uv()uv[(
3/)cabaa(a
+++++==++=
+++==++=
++++==++=
for Fermats equations : nnn cba =+
With all this going on, triads ooo c,b,a generate :
1. A five forms
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for separation endless series prime numbers, see below Part 2:
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(Aabif)ab(A
baif)ba(A
abif)ab(A
=>=>=>=>=
2. A non-modular elliptical curves. For this curves is special variant Freys-Yaroshs equations:
2o
2o
2o
2
cba
)BX(X)AX(Y
==+=
and general variant equations:
no
no
no
2n
cba
)BX(X)AX(Y
==+=
where
no
2o
no
2o
aBoraB
bXorbX
====
3. A orthogonal coordinates for complex numbers:
oo
oo
biasbias=+=
and for complex function (complex invariant):
2222o
2o
2o
oooo
)uv(cba
)bia()bia(ssS
+==+==+==
4. A two zeta functions: Riemanns zeta function
0)S1()S1(2Ssin2)S( 1SS =
and autors zeta function:
==n
SQsin2)1()S( s
a key to finally proofs a conjectures: of Riemanns, of Birch and Swinnerton-Dyers
5. A statement:
Hypothesis of Shimura-Taniyama All elliptic curves are modular curve,[1],
is erroneous hypothesis . According, reasoning by doctor A.Wiles, [2] , is faulty reasoning.
As alternative and as confirmatory evidence this fact, autor to make an offer application Abel group and
Diophantine equations as universal mathematical formulation for proofs
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Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and Swinnerton-Dyers.
PACS numbers: 02.10.Ab 02.30.Xx
T a b l e o f c o n t e n t s
Instead of Introduction Part 1, Yuri Zhivotov , see [5],
against argumentations by Ken Ribet and Andrew Wiles . A u t o rs P r o p o s a l s Part 2 , History
Part 3 , Possible variants proof of Fermats Last Theorem over Q and applications Abel group. Alternative Wiles proof
Part 4, Non-modular elliptic curves, 10-th problem of D.Hilbert, as way to proof of Riemann Hypothesis
Part 5 , Intercommunication between the elliptical curves, Abel group and the non-modular forms.
Part 6, Riemanns sphere as mapping of space task common solutions of Conjecture
Birch and Swinnerton-Dyer and Riemann Hypothesis
Part 7 , Proof Riemanns Hypotesis Part 8 ,C o m m e n t a r y
to the question of Dyophantines equations and their irrational roots ,containing
information of non-modular elliptical curves. Part 9, Proof of Conjecture Beal Reference
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Instead of Introduction
SYSTEMS A COORDINATS AND GENERAL INVARIANT
System A coordinats for whole primitive Pythagoras numbers
Pic.1
This is ortogonal system coordinate for primitive
Pythagorean triads:
22
0
0
220
uvcvu2b
uva
+==
=
as for all pair v>u numbers are the numbers
of various evenness taken from endless series , see (3) :
,...4,3,2,1)1u(v,....,4,3,2,1,0u
=+==
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C o n s e q u e n c e 1
2o
o2o
2o
oo
2
ucu2
buav
vcv2
bavu
==+=
=== (4)
System B coordinates for my complex numbers
Pic. 2
Geometrical interpretation a vectorial product
Pic.3
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Here we create a product:
220
20
0000
ba
)bia()bia(ssS
=+==+==
(5)
It is general common mathematical invariant for systems A and B AS COMPLEX FUNCTION
over field N of natural numbers uv >
Other, we create a vectorial product:
[ ]s,s (6) and vectors length :
Qsinsss,sS == (7) as argument for equations:
0)S1()S1(2Ssin2)S( 1SS == (8)
GENERAL MATHEMATICAL STRATEGICS A SOLUTION OF THE BIRCH AND SWINNERTON-DYER CONJECTURE
Defining role in the proof by Birch and Swinnerton-Dyer belongs to
the EXPANDED or CLOSED plane of complex numbers: += iz and = iz
properties of this plane are defined by properties of Riemanns Sphere below, on Pic.4 Riemanns sphere [3] is represented.
In a Fig. 5 it is presented diametrical section of Riemanns sphere
Here : Flatness ),( for complex numbers += iz and = i'z ;
Point )b,a(PP oo= ; Point )Y,,('P'P = ; Axis absciss for ; Axis ordinat for ;
It is stereographycal projection flatness ),( to sphere )Y,,(
Pic. 4
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Stereographycal projection keep angles . thats have mathematical description, see (32):
2o
2o
2o
2o
2o
2o
2o
o2
o2
o
o
20
02
o2
0
o
c1c
ba1bay
c1b
ba1b
c1a
ba1a
+=+++=
+=++=+=++=
In a Fig. 5 it is presented diametrical section of Riemans sphere
Pic.5
INITIAL DATA
V A R I A N T D A T A 1
If angle =Q , see Pic.2 and Pic.3, then:
0)s(Re0)s(Re
ibib
o
o
==++
(9)
According we have endless series numbers:
2o
oooo
ibaibassS
==+==
(10)
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as endless series primitive numbers of Pythagora over field N natural numbers )uv( > :
)uv(S 22o +== (11) In the end we have endless products,
as endless series seros:
00sinsss,sS === (12) According this statement, we create endless series
functions for proof a Birch and Swinnerton-Dyer conjecture
0)S1()S1(2Ssin2)S( 1SS == (13)
Here scalars 0S = are arguments for theyre functions .
Pic. 6
The axis of absciss oa is geometrical axis symmetry for two a system coordinat.
Common the angle Q is general argument for general functions: 0)S1()S1(
2Ssin2)S( 1SS == (14)
V A R I A N T D A T A 2
If angle 0Q = , see Pic. 3, then:
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o
o
o
o
a)s(Rea)s(Re
0ib0ib
==
==+
(15)
According we have endless series numbers:
20
0000
a
)bia()bia(ssS
==+==
(16)
as endless series primitive numbers of Pythagora over field N natural numbers:
)uv(Sa 22o == (17) In the end we have endless vectorials products, as endless series sero:
00sinss]s,s[S === (18) According this statement, we create endless series functions
for proof a Birch and Swinnerton-Dyer conjecture:
0)S1()S1(2Ssin2)S( 1SS == (19)
Here scalars 0S = are arguments for theyre equations.
0bi+
oib Pic. 7
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Arithmetic of modular elliptic curves
and diophantine equations
Basis of arithmetic of modular elliptic curves contains five statements.
Statement 1
Let 1N whole number, and let )N(0 is multitude all matrix:
2221
1211
aaaa
)22(M = Where )a;a;a;a( 22211211 whole numbers , N divide 21a and :
1aaaa
)22(DDet
12212211
===
=
Statement 2
(Mazur)
A Frey elliptic curve: )BX(X)AX(Y2 +=
has no Q-rational subgroup of prime order 2n The inexistence of Q-rational points of prime
order 2n on Frey curves is sufficiently.
Statement 3
Hypothesis of Shymura-Taniyama All elliptic curves are modular curve,[1]
is sufficiently
Statement 4
A.Wiles proved: Hypothesis of Shymura-Taniyama equitable for a Frey elliptic curve.
Statement 5
If Fermats Last Theorem is proved for n=4, there is no need
of prove it for all even exponents of degree for Fermat equation Autors [3] and [4] it is claimed that :
Suffice it to prove for n=4 and for n=p . Here p arbitrary value of prime numbers.
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Arithmetic of non-modular elliptic curves and diophantine equations
Basis of arithmetic of non-modular elliptic curves
contains five counterstatement.
Counterstatement 1
Let 1N whole number, and let )N( is multitude all matrix:
333231
232221
131211
aaaaaaaaa
)33(M =
where
)a;a;a;a;a;a;a;a;a( 333231232221131211
whole numbers , and :
0)33(MDet =
Equivalent matrix )33(M :
===
===
===
=
no33
2no
2o32
2no
2o31
2no
2o23
no22
2no
2o21
2no
2o13
2no
2o12
no11
cabcaaca
cbabaaba
caabaaaa
)33(M
Matrix )33(M based at the primitive triads of Pythagorean.
Key condition for primitive Pythagorean triplet. Diophantus and then Fibonacci , indicated the following method
of search of solutions of Pythagorean equation :
If whole numbers v and u are such that v > u and greatest common divisor GCD (v, u)=1 , at that v and u
of different evenness, than triads ooo c,b,a , generate a endless series of diophantines equations:
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12
220
0
220
uvcuv2buva
+==
=
as endless series a primitive solutions for Pythagorean equations:
2o
2o
2o cba =+
and as endless series irrational roots:
n n222n2222n22222
n no
2no
2o
2no
2o
n 2n222n2n222
n 2no
2o
no
2n2o
n 2n222222n22n22
n 2no
2o
2no
2o
no
3/])uv()uv2()uv()uv()uv[(
3/)cbcac(c
3/])uv()uv2()uv2()uv()uv2[(
3/)cbbab(b
3/])uv()uv()uv2()uv()uv[(
3/)cabaa(a
+++++==++=
+++==++=
++++==++=
for Fermats equations : nnn cba =+
with all this going on, triads ooo c,b,a generate :
1. A five forms for separation endless series prime numbers,
see below Part 2:
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(Aabif)ab(A
baif)ba(A
abif)ab(A
=>=>=>=>=
2. A non-modular elliptical curves. For this curves is special variant Freys-Yaroshs equations:
2o
2o
2o
2
cba
)BX(X)AX(Y
==+=
and general variant equations:
no
no
no
2n
cba
)BX(X)AX(Y
==+=
where
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no
2o
no
2o
aBoraB
bXorbX
====
3. A orthogonal coordinates for complex numbers:
oo
oo
biasbias=+=
and for complex function (complex invariant):
2222o
2o
2o
oooo
)uv(cba
)bia()bia(ssS
+==+==+==
4. A two zeta functions: Riemanns zeta function
0)S1()S1(2Ssin2)S( 1SS =
and autors zeta function:
==n
SQsin2)1()S( s
a key to finally proofs a conjectures: of Riemanns, of Birch and Swinnerton-Dyers
5. A statement:
Hypothesis of Shimura-Taniyama All elliptic curves are modular curve,[1],
is erroneous hypothesis . According, reasoning by doctor A.Wiles, [2] , is faulty reasoning.
As alternative and as confirmatory evidence this fact, autor to make an offer application Abel group and
Diophantine equations as universal mathematical formulation for proofs
Fermats Last Theorem and Conjectures: Riemanns, Beals, Birchs and Swinnerton-Dyers.
Here pair natural numbers
uv > as basis for solutions common Problems
o f F e r m a t , Beal , R i e m a n n , B i r c h a n d S w i n n e r t o n D y e r
Whole numbers ooo c,b,a is abelian varieties over Q
and over field natural numbers uv >
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Counterstatement 2
A Frey elliptic curve : )BX(X)AX(Y2 +=
has Q-rational subgroup of prime order 2n The existence of Q-rational points of prime order 2n on Frey curves is sufficiently.
Counterstatement 3
Hypothesis of Shymura-Taniyama All elliptic curves are modular curve,
is faulty Hypothesis.
Counterstatement 4
A.Wiles proved: Hypothesis of Shymura-Taniyama equitable for a Frey elliptic curve
as semistable modular elliptic curves. It is sufficiently. But there is just one snag (to it):
Exist my non-modular elliptic curve. For this curve is special variant Freys-Yarosh equations:
2o
2o
2o
2
cba
)BX(X)AX(Y
==+=
and general variant equations
no
no
no
2n
cba
)BX(X)AX(Y
==+=
where
no
2o
no
2o
aBoraB
bXorbX
====
and )ab(Aor)ab(A no
no
2o
2o == conductor-controller.
Instead a formula for selection prime numbers, see [1]:
pp 1pa += where p is prime numbers and p
is analog for numbers:
+=
)4(mod1pfor1p)4(mod1pfor1p
Np
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I propose three Rule for computations all prime numbers iA
Because:
Exist five forms
for separation endless series prime numbers, see below Part 2:
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(Aabif)ab(A
baif)ba(A
abif)ab(A
=>=>=>=>=
Here every number from endless series prime numbers
)ab(A 202
01 = if oo ab > will not divide into
222
22
0
0
)()2(
)122b(4)vu2b(416
=======
If number
2
n4ooo )cba(16
=
is function a discriminant my elliptic curves:
8
n2ooo
2)cba( =
At the same time )ab(A 2o
2o = will not divide
3
22o
220
3)12a(9
)uva(927
====
===
if odd number 27 is equivalent:
+= 2
o
3o
ba427
where )b27a4( 2o
3o +=
discriminant for canonical form any elliptic curves:
oo32 bxaxY ++=
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Endless series prime numbers:
002
02
01 abif)ab(A >= is endless series of conductors-criterions
for separation non- modular elliptic curves. Number 2 end number 3 were invariants
from endless series prime numbers. All prime numbers settle down in a natural line in pairs,
intervals between which submit to a rhythm of numbers v=2 and 3a0 = :
2
2
2131721113
2711257235123112
=======
2
2
3
2
243472414323741
323137223312192321719
===
====
27173263712616325761253572515324751
2
2
2
2
=======
2
2
2
2
2
2971012939728993287892838727983
327379
======
=
......................2117121211311721111132109111210710921031072101103
2
2
2
=======
All differences between the next simple numbers are subordinated to the law of formation of primes-numbers and spectral invariant:
2)2...2...2222( 32102 +++++++=
Following spectral forms contains this invariant:
21ii21ii )AA(or)AA( == 21ii 2)AA( = 21ii 3)AA( = 21ii 4)AA( =
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Note: AandA 1
are isogenous abelian varieties
Counterstatement 5
If Fermats Last Theorem is proved for n=4, there is no need of prove it for all even exponents of degree for Fermat equation
Autors [3 ] and [4] it is claimed that : Suffice it to prove for n=4 and for n=p . Here p arbitrary value of prime numbers.
It is faulty principle.
Explanatory example for Counterstatement 5
Initial data
If 4n = , then common multiplier:
3503/)cba(D 2n0
2n0
2n0n
4 =++= If 8n = , then common multiplier :
3450203/)cba(D 2n0
2n0
2n0n
8 =++= Here primitive Pythagorean triplet:
5uvc4uv2b3uva
220
0
220
=+=====
and basis
1u2v
==
Statement:
If equation:
444 cba =+ have roots:
...511801.4c...041031.4b...996355.3a
4
4
4
===
then roots:
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...507634.4c...2629624.4b...9671335.3a
8
8
8
===
for equation
888 cba =+ is independent roots .
Because:
...]507634.4c[]...)511801.4()c[(...]2629624.4b[]...)041031.4()b[(...]9671335.3a[]...)996355.3()a[(
8224
8224
8224
======
With all this gong on two triads irrational value roots For Fermats equation:
nnn cba =+
First triad for degree 4n = :
...499635512.33
1503
509Daa 44n n42
04 ====
...041031009.43
8003
5016Dbb 44n n42
04 ====
...518010018.43
12503
5025Dcc 44n n42
04 ====
Second triad for degree 8n = :
...96713355.33
1840503
204509Daa 888 n82
08 ====
...262962429.43
3272003
2045016Dbb 888 n82
08 ====
...507533969.43
5112503
2045025Dcc 888 n82
08 ====
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GENERAL RESULT:
...]507533969.4c[]...)518010018.4()c[(...]262962429.4b[]...)041031009.4()b[(
...]96713355.3a[]...)499635512.3()a[(
8224
8224
8224
======
C O M M E N T
Two the way work out a problem of P.Fermat: 1. Deductive (intuitive) way
2. Inductive way
1. Deductive way
Let 888 zyx =+ equation of P. Fermat A priori it is known:
...507533969,4z...262962429.4y...967133355.3x
===
Issue:
How did I do it ?
Answer:
Enigma
3. Inductive way
Let Cz,By,Ax nn === whole or rational numbers. Then:
n
n
n
cz
By
Ax
===
roots for equation of P.Fermat: nnn zyx =+
If CBA =+
where vectors
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)cba(cDcC
)cba(bDbB
)cba(aDaA
2no
2no
2no
2oN
2o
2no
2no
2n2ob
2o
2no
2no
2no
2on
2o
++==++==++==
composed of primitive Pythagorean triplets.
220
0
220
uvcuv2buva
+==
=
4. Basis for Inductive way is Abelian group of three whole numbers:
cCvectorsorccsquares
bBvectorsorbbsquares
aAvectorsoraasquares
2o
2o
2o
Abelian group of whole numbers a,b,c (from Internet)
I chose to begin with the notes out of which I constructed the central definition below. The equation which defines distributivity is:
a(b+c) = ab + ac
This has, of course, a `reversed' form, (b+c)a = ba+ca: I chose to name the displayed form `left' distributive and this latter form `right' distributive. When cast in the general terms of binary operators, naming multiplication f and addition g, we have, for any legitimate a, b
and c:
f(a, g(b,c)) = g(f(a,b), f(a,c))
Thus, if we take (AB|f:C) and (DE|g:F) as temporary namings for the domains and ranges of our binary operators, we obtain
a is in A; g(b,c) (in F), b and c are in B; f(a,b) (in C) and b are in D; and f(a,c) (in C) and c are in E.
so we need F to be a subset of B and C to be a subset of D and of E. I chose to take C=D=E=F=B for this left-distributive case, replacing B with A for right-distributive.
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Distributivity
A binary operator, (AB|f:B), left-distributes over a uniform binary operator, g, on B precisely if, for every a in A and b, c in B: f(a,g(b,c)) = g(f(a,b),f(a,c)). We say (BA|f:B) right-distributes over (BB|g:B) precisely if, for every a in A and b, c in B: f(g(b,c),a) =
g(f(b,a),f(c,a)). One binary operator is said to distribute over another precisely if the former both left-distributes and right-distributes over the latter - in which case both are necessarily
uniform and the two are parallel (that is, they act on the same space).
In particular, any Abelian binary operator which left- or right-distributes over some binary operator inevitably distributes over the latter. When B and A are distinct, (AB|f:B) can
only distribute from the left over anything, and that must be over some (BB|:B), so there is no ambiguity in refering to such an f as distributing over some g, implicitly uniform on B. It should also be noted that if f does left-distribute over some g, then its transpose, (BA| (b,a)-
>f(a,b) :B), right-distributes over g.
Further reading
An (AB|:B) may left-distribute over a (BB|:B): compare and contrast with an (AA|:|) left-associating over an (AB|:B). The combination of these forms the cornerstone of the
notion of linearity, which underlies such fundamental tools as scalars and vectors.
P A R T 1
Yuri Zhivotov , see [5], against
argumentations by Ken Ribet and Andrew Wiles.
A u t o r s P r o p o s a l s
Where is the Logic of Great Fermat's Theorem Proof?
The analysis made shows that one should not believe Singh's book. This is a literary
work. So, there should be a General proof of the Great Fermat's theorem. There is not any. Some people say that Frey established a link between the Fermat's theorem and Taniyama-
Shimura's hypothesis. The others assert that Frey assumed that the proof of Taniyama-Shimura's hypothesis would automatically prove the Great Fermat's theorem. But Frey's article is inaccessible for a reader. The third assert that Ribet proved Frey's assumption.
The forth consider that Ribet proved that Frey's curve was not modular. The fifth consider that Taniyama-Shimura's hypothesis was proved by Wiles, and so on. Moreover, the
assertions of some people contradict to the assertions of the others. Everybody admires the proof of Great Fermat's theorem but nobody saw it in full scope. And taking into account the proofs of Fermat's theorem for the cases n=3, n=4. The riddle of a proof. The search in
the Internet did not help to solve the riddle. The result is very unexpected and pitiful. However there are three personages in this riddle: Gerhard Frey, Ken Ribet, Andrew
Wiles. Evidently, one should look into the works of these mathematicians. Perhaps then it
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will be possible to understand how the Great Fermat's theorem was proved. Let us believe that none of the mathematicians, except Frey, made errors (Singh informs on his errors). If
any suspicions arise that additional errors exist, we will substantiate them.
The main mistake of Andrew Wiles it the fact that he got involved with the proof of Great Fermat's theorem. All the more, the mathematicians should not have made a mistake
collectively. Let us trace the way from Taniyama-Shimura's hypothesis
to Fermat's theorem, of course, if we manage it. Taniyama-Shimura's hypothesis states that any elliptical curve is modular.
In particular, the elliptical curve described by the equation: )DX()KX(XY2 += (1)
with the integer coefficients must be modular. Andrew Wiles proved Taniyama-Shimura's hypothesis.
That is he proved that the elliptical curve described by the equation: )DX()KX(XY2 += (1)
with the integer coefficients was modular. That is the following equations correspond to modular curves:
)5X()3X(XY2 += (2)
)25X()9X(XY2 += (3)
)125()27X(XY2 = (4)
)625X()81X(XY2 += (5)
)3125X()243X(XY2 += (6)
Equation (6) may be written in the form:
)5X()3X(XY 552 += (7)
Or )BX()AX(XY nn2 += (8)
Ken Ribet proved that the elliptical curve
described by the equation
)BX()AX(XY nn2 += (9)
was not modular. Ken Ribet contradicts to Andrew Wiles. This is a deadlock. Perhaps, somebody wants to say that the numbers nA and nB do not exist?
Perhaps, somebody wants to say that the numbers nA and nB are included into the hypothetical Fermat's equation
nnn CBA =+ (10)
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and therefore the numbers nA and nB do not exist?
Everybody has the right to make assumptions but assumptions must be proved.
Besides, it should be taken into account that in the equations (2)-(6) also have an uncertainty.
The equations can be put into another form (in an analogous way equation (6) was modified into equation (7)).
But the theory of elliptical curves does not know how to determine the discriminant of the equations.
It is strange that the discriminant can be found for equation (7).
3.8 At one of the steps of his proof Ribet proposes to pay attention to the minimal discriminant of Frey's curve (as it appeared, existent, semistable and modular).
Ribet proposes to write the minimal discriminant of Frey's curve in the following form:
8n2 2/)cba(d = (11)
Pay attention. If number "c" does not exist, the discriminant does not exist. If the discriminant does not exist, the Frey's curve does not exist.
If the Frey's curve does not exist, the curve is fictitious. Just another time Ribet tries to divert a reader's attention
from the real existence of Frey's curve. Let us remind a reader that the discriminant of elliptic curve has the form:
2nnnn )]ba(ba[16D += (12)
Therefore, the minimal discriminant can be written in the form:
82nnnn 2/)]ba(ba[d += (13)
or 82nn 2/)]C(ba[d = (14)
Such record of the minimal discriminant excludes the doubt about existence of Frey's
curve, which is semistable and modular. Such record of the minimal discriminant reduces the idea of Fermat's theorem proof to the consideration of possibility of minimal
discriminant factoring, that is number c presentation in the form of exponential number nc . In this case the Ribet's exercises with putting the Fermat's numbers
into the Frey's elliptic curve equation are not needed. However, as it follows from the article, Ribet made every effort to conceal this
simple truth and to substitute it with the reasonings about the link between Frey's curve and Fermat's equation.
Let us consider the reasons, for which Ribet forcedly conceals the truth.
-
24
Conclusions
Ribet did not prove Fermat's theorem in the assumption of the truth of Taniyama-Shimura hypothesis.
Ribet made too many mistakes and discrepancies, which allows to consider his proof as an unsuccessful attempt.
The existence of Bill's conjecture also strikes a blow at Ribet's proof, which obviously is impossible to ward off.
Let me express perplexity to Andrew Wiles, which is connected with the use of Ribet's work in the "general" proof of Fermat's theorem,
to which Wiles has pretensions. So many mistakes were revealed in several lines of the proof that it is hard to believe that they had not
been noticed by the specialist in this field of mathematics. The Great Fermat's theorem is connected with
the history of mathematics, and it is impermissible to treat it haughtily.
A u t o rs P r o p o s a l s
Autor of these article proposes to write the minimal discriminant
of Frey's-Yaroshs non-modular elliptic curve:
2o
2o
2o
2
cba
)BX(X)AX(Y
==+=
(15)
where
2
o
2o
aB
bX
==
(16)
And )ab(A 2o
2o = (17)
in the following form:
02
)]uv()uv2[(2
]uv()uv2()uv[(2
)cba(
8
n244
8
n22222
8
n2ooomin
=
+=
==
(18)
If v > u are the numbers of various evenness taken from endless series of natural numbers , vide supra, then :
-
25
n
o2n
o2
o2n
o2
on
2no
2o
no
2no
2o
n
2no
2o
2no
2o
no
n
cbcacC
cbbabB
cabaaA
++=++=++=
(19)
three vectors over field Q natural numbers.
It is objects to Abel group f,G with binary dealership: Action associative
oooooo
oooooo
ooooo0
b)ac(baca)cb(acbc)ba(cba
===
(20)
Left distributive
ooooooo caba)cb(a +=+ (21) Right distributive
ooooooo acaba)cb( +=+ (22) The combination of this form the cornerstone of the motion of linearity,
which underlies such fundamental tools as scalars and vectors
n
o2n
o2
o2n
o2
on
2no
2o
no
2no
2o
n
2no
2o
2no
2o
no
n
cbcacC
cbbabB
cabaaA
++=++=++=
(23)
With all this going on:
n
on1
o
no
n1o
no
n1o
c)c(
b)b(
a)a(
===
(24)
1Ecba 0o0
o0
o ==== (25)
mn
om
on
o
mno
mo
no
mno
mo
no
ccc
bbb
aaa
+
+
+
===
(26)
n
mno
mno
mno D3)cba( =++ (27)
where
-
26
3/)cba(D mnomn
omn
on ++= (28)
Consequence
Linear combination
n n
o2n
o2
o2n
o2
on n
n 2no
2o
no
2n2o
n n
n 2no
2o
2no
2o
no
n n
3/)cbcac(CCc
3/)cbbab(BBb
3/)cabaa(AAa
++===++===++===
(29)
are endless series roots to Fermats equations :
nnn CBA =+ (30)
Here and everywhere:
22
0
0
220
uvcuv2b
uva
+==
= (31)
primitive Pythagorean triplets
I did consider all cases that appear at the "introduction" of the numbers
nA ,
nB and nC (32)
from Fermat's equation:
nnn CBA =+ (33)
into general the Frays-Yarosh
non-modular elliptic curve equation:
no
no
no
2
cba
)BX(X)AX(Y
==+= (34
if vectors
-
27
nno
2no
2o
2no
2o
n
n2no
2o
no
2no
2o
n
n2no
2o
2no
2o
no
n
CcbcacC
BcbbabB
AcabaaA
++=++=++=
(35)
generate whole numbers as linear combinations
)bcac(Cc
)cbab(Bb
)caba(Aa
2no
2o
2no
2o
nno
2no
2o
2no
2o
nno
2no
2o
2no
2o
nn
+=+=+=
(36)
In this case
linear combination generate three multitude irrational numbers
n n
o2n
o2
o2n
o2
on n
n 2no
2o
no
2n2o
n n
n 2no
2o
2no
2o
no
n n
3/)cbcac(CCc
3/)cbbab(BBb
3/)cabaa(AAa
++===++===++===
(37)
as endless series odd numbers or as roots to Fermat equation:
nnn CBA =+ (38)
Pay attention.
If whole primitive Pythagorean numbers
220
0
220
uvcuv2b
uva
+==
= (39)
does exist, then the discriminant
02
)]uv()uv2[(2
]uv()uv2()uv[(2
)cba(
8
n244
8
n22222
8
n2ooomin
=
+=
==
(40)
also does exist.
-
28
If the discriminant does exist, then the Frey's Yaroshs
non-modular elliptic curve also does exist.
D E D U C I N G
Hypothesis of Shimura-Taniyama All elliptic curves are modular curve,[2],
is erroneous hypothesis . According, case- based reasoning by doctor A.Wiles, [3] , is erroneous either.
P A R T 2 History
In mathematics, the modularity theorem establishes an important connection, between elliptic curves over the field of rational numbers and modular forms, certain analytic
functions introduced in 19th century mathematics. It was proved, for all elliptic curves over the rationals whose conductor (see definition below) was not a multiple of 27, in
fundamental work of Andrew Wiles and Richard Taylor. The result had previously been called the TaniyamaShimuraWeil conjecture, or related names. The great interest in the
theorem was that it was already known to imply Fermat's Last Theorem, a celebrated unsolved problem on diophantine equations.
The remaining cases of the modularity theorem (of elliptic curve not with semistable reduction) were subsequently settled by Christophe Breuil, Brian Conrad, Fred Diamond,
and Richard Taylor .
An incorrect version of this theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama died in 1958. The conjecture was rediscovered by Andr Weil in 1967, who showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. In the 1970s it became associated with the Langlands program of unifying conjectures in mathematics.
It attracted considerable interest in the 1980s when Gerhard Frey suggested that the TaniyamaShimuraWeil conjecture implies Fermat's last theorem. He did this by
attempting to show that any counterexample to Fermat's last theorem would give rise to a non-modular elliptic curve. Ken Ribet later proved this result. In 1995, Andrew Wiles, with
the partial help of Richard Taylor, proved the modularity theorem for semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.
The full modularity theorem was finally proved in 1999 by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases
until the full result was proved.
Several theorems in number theory similar to Fermat's last theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n-th
powers, n 3. (The case n = 3 was already known by Euler.)
-
29
References
Henri Darmon: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.
Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor: On the modularity of elliptic curves over Q: Wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), pp. 843939. Contains the proof of the modularity theorem.
Barry Mazur, Number theory as gadfly- American Mathematical Monthly, 98 (7), August-September 1991, pp. 593610, Disscusses the Taniyama-Shimura conjecture 3 years before it was proven for infinitely many cases.
Weil, Andr ber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168 1967 149-156.
Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443--551.
Taylor, Richard; Wiles, Andrew Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), no. 3, 553--572
Wiles, Andrew Modular forms, elliptic curves, and Fermat's last theorem. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zrich, 1994), 243--245, Birkhuser, Basel, 1995.
Retrieved from "http://en.wikipedia.org/wiki/Modularity_theorem"
-
30
P A R T 3
Possible variants proof of Fermats Last Theorem over Q
and applications Abel group, as alternative Wiles proof
Alternative statement
I offer to you attention a solution Fermats problem as solution of system of equations P. Fermat.
The are enough reasons to assume that P. Fermat
considered the equation:
nnn cba =+ (41)
-
31
as an infinite system of independent equations with finite number of variable n a t u r a l numbers (a, b, c, n) :
..................
..................cba
.................
.................cbacbacba
nnn
444
333
222
=+
=+=+=+
(42)
He knew that the equation (1) at n = 2 came from extreme antiquity. It has geometrical interpretation
and is called the equation of Pythagoras (approx. 580 500 B.C.):
222 cba =+ (43)
It was also known that among the infinity aggregate of solutions of equation (3) there are such threes
of numbers )c,b,a( 000 that do not have common multipliers. These threes of
n a t u r a l numbers are known as p r i m i t i v e t h r e e s of Pythagorean. These primitive threes of Pythagoras
were also known to be generated by any couple v > u of natural numbers
of different parity with the help of three invariant forms:
22
0
0
220
uvcuv2b
uva
+==
= (44)
-
32
The further line of argument is obvious.
Four variable natural numbers )n,c,b,a( 00o are building three forms:
n n2
0 Daa = (45) n n
20 Dbb = (46)
n n2
0 Dcc = (47) Here
3/)cba(D 2n02n
02n
0n ++= (48)
universal common multiplier
Key condition for primitive Pythagorean triplet: Diophantus and then Fibonacci , indicated the following method
of search of solutions of Pythagorean equation :
222 zyx =+ (49)
If whole numbers v and u are such that v > u and
greatest common divisor GCD (v, u)=1 , at that v and u of different evenness than triads ooo c,b,a , given by equations:
22
0
0
220
uvc
uv2buva
+==
= (50)
are primitive solutions of Pythagorean equation :
2o
2o
2o cba =+ (51)
N O T E Pair natural numbers
uv >
I s b a s I s f o r P r o b l e m s o f F e r m a t , R i e m a n n , B i r c hs a n d S w i n n e r t o ns D y e rs
See Diagram 1 and Plan 1
-
33
Diagram 1
All prime numbers settle down in a natural line in pairs,
intervals between which submit to a rhythm of numbers v=2 and 3ao = :
2
2
2131721113
2711257235123112
=======
2
2
3
2
243472414323741
323137223312192321719
===
====
27173263712616325761253572515324751
2
2
2
2
=======
2
2
2
2
2
2971012939728993287892838727983
327379
======
=
......................2117121211311721111132109111210710921031072101103
2
2
2
=======
(52)
Et cetera, et cetera
-
34
G e n e r a l c o n c l u s i o n
See formerly :
Basis for Inductive way is Abelian group of three whole numbers:
cCvectororccsquare
bBvectororbbsquare
aAvectororaasquare
2o
2o
2o
General basis for all rightly solutions to problems of Last theorem, Riemanns conjecture,
Birch and Swinnerton-Dyer conjecture is Abel Group f,G .
C o n s e q u e n c e 1
If v > u are the numbers of various evenness taken from endless series of natural numbers , vide supra, then :
n
o2n
o2
o2n
o2
on
2no
2o
no
2no
2o
n
2no
2o
2no
2o
no
n
cbcacC
cbbabB
cabaaA
++=++=++=
(53)
Three vectors over field Q natural numbers.
It is objects to Abel group f,G with binary dealership: Action associative
oooooo
oooooo
ooooo0
b)ac(baca)cb(acbc)ba(cba
===
(54)
Left -distributive
ooooooo caba)cb(a +=+ (55)
-
35
Right- distributive
ooooooo acaba)cb( +=+ (56)
The combination of this form the cornerstone of the motion of linearity, which underlies such fundamental tools as scalars and vectors
n
o2n
o2
o2n
o2
on
2no
2o
no
2no
2o
n
2no
2o
2no
2o
no
n
cbcacC
cbbabB
cabaaA
++=++=++=
(57)
with all this going on:
n
on1
o
no
n1o
no
n1o
c)c(
b)b(
a)a(
===
(58)
1Ecba 0o0
o0
o ==== (59)
mn
om
on
o
mno
mo
no
mno
mo
no
ccc
bbb
aaa
+
+
+
===
(60)
Pay attention.
Everywhere
n
mno
mno
mno D3)cba( =++ (61)
where
3/)cba(D mnomn
omn
on ++= (62)
-
36
Consequence 2
Linear combination
n n
o2n
o2
o2n
o2
on n
n 2no
2o
no
2n2o
n n
n 2no
2o
2no
2o
no
n n
3/)cbcac(CCc
3/)cbbab(BBb
3/)cabaa(AAa
++===++===++===
(63)
endless series roots to Fermats equations :
nnn CBA =+ (64)
P A R T 4
Non-modular elliptic curves,
10-th problem of D.Hilbert, and Rule A , Rule B , Rule C
as way to proof of Riemann Hypothesis
I propose:
1. Co-prime bases for zyx CBA =+ as bases for irrational roots by Fermats equation:
nnn cba =+ (65)
2. Instead of parity p by Frey:
)p(mod)bX()pX(XY qq2 + (66)
deterministic expression
)BX(X)AX(Y2 += (67)
In this case we have substitutions:
-
37
2
0
2o
20
c)BX(
bX
a)AX(
=+=
=
(68)
where
2o
2o
2o
aB
abA
==
(69)
and
)uv(c
)vu2(b)uv(a
220
0
220
+==
= (70)
primitive Pythagorean triplets for all
uv > (71)
are the numbers of various evenness , taken from endless series of natural numbers.
Here every number from endless series prime numbers
)ab(A 202
01 = if oo ab > (72)
will not divide into
22
222
0
0
)()2(
)122b(4)vu2b(416
=======
(73)
if number
2n4
ooo )cba(16 = (74)
is function a discriminant my elliptic curves , see (15) and (34):
8n2
ooo
2)cba( = (75)
At the same time )ab(A 20
201 = will not divide
3
22o
220
3)12a(9
)uva(927
====
=== (76)
-
38
if odd number 27 is equivalent:
+= 2
o
3o
ba427 (77)
where )b27a4( 2o
3o += (78)
discriminant for canonical form any elliptic curves: oo
32 bxaxY ++= (79) Endless series prime numbers:
002
02
01 abif)ab(A >= (80) is endless series of conductors-criterions
for separation non- modular elliptic curves.
As result I receive my equation for endless series to non-modular elliptic curves for 2n = :
2
o2
o2
o2 cbaY = (81)
and for 2n>
no
no
no
n cbaY = (82) In this case:
n
0
no
n0
c)BX(
bX
a)AX(
=+=
=
(83)
where
no
no
no
aB
abA
==
(84)
and
)uv(c
)vu2(b)uv(a
220
0
220
+==
= (85)
primitive Pythagorean triplets for all
uv > (86)
are the numbers of various evenness , taken from endless series of natural numbers.
Here every number from endless series numbers
)ab(A n0n
0 = (87) will not divide into
-
39
22
222
0
0
)()2(
)122b(4)vu2b(416
=======
(88)
and will not divide
333927 == (89)
In the end, if
5c4b3a
o
o
o
===
(90)
then we have a minimal discriminant for 2n = :
62550
2)543(2
)cba(
8
4
8
n2ooo
==
== (91)
where
0 (92)
As explanatory, for 15n = :
53
8
30
8
n2ooo
102739210.22
)543(2
)cba(
==
== (93)
Consequently, my curve is non-singular, my curve is elliptical curve.
2. Instead of Freys formula:
pp v1pa += (94)
I propose three of Rule for computation all simple (prime) numbers :
Rule A, Rule B and Rule C.
It is original and novel results analysis of knowledge data about natural and primes-numbers, about primitive triplets of Pythagorean, about equations of Pythagorean,
-
40
P.Fermats and G.Freys equations. General result three Rules: Rule A, Rule B and Rule for separate and calculate endless series prime-numbers. It is adequate
interpretation conclusion of Matiyasevich, who has proved 10-th problem of D.Hilbert, proves to be true:
All prime numbers are simple search (recalculation) of all of some natural numbers.
Three forms of Rules, in according to conception by D.Hilbert, [12] and [11], creates expansion endless series forms, which a useful to become of fractals, see forms (21)(24),
end primes-numbers. The forms contains spectral invariant 2 , invariant )5(R5
= ,
invariant )7(
R7
= and general invariant complex function
20
20
0000
ba
)bia()bia(ssS
+==+==
(95)
as basis for calculations of gtneral zeta function:
0)S1()S1(2Ssin2)S( 1SS = (96)
and my zeta function:
==n
SQsin2)1()S( s (97-8)
Here Q variable quantity of angle, see Pic.1 and Pic.2 : Q0 (98-9)
and ...,5,4,3,2,1n=
Remark
G.F.B. Riemann (1826 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function
...4/43/12/11)s( 555 +++++= called the Riemann Zeta function - see: http://www.claymath.org/millennium.)
Primitive Pythagorean triads ,which halt a value primes numbers by a rule A
R U L E A
A ) If 50 Ac
= primesnumbers , then according :
-
41
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(Aabif)ab(A
baif)ba(A
abif)ab(A
=>=>=>=>=
(99)
For 00 ab > , numbers
)ab(A 003 = (100)
are primesnumbers and
)ab(A 202
01 = (101)
are primesnumbers or
)ab( 202
0 (102)
will divide by controller
)7(R7
= (103)
For 00 ba > , numbers
)ba(A 004 = (104)
are primesnumbers and
)ba(A 202
02 = (105)
are primesnumbers or
)ab( 202
0 (106)
will divide by controller
)7(R7
= (107)
-
42
Primitive Pythagorean triads, which halt a value primes numbers by a rule B
R U L E B
If controller
)5(
R5
= (108) divide numbers 0c , then according :
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(A
abif)ab(Abaif)ba(A
abif)ab(A
=>=>=>=>=
(109)
For 00 ab > , numbers
)ab(A 003 = (110) )ab(A 20
201 = (111)
are primsnumbers
For 00 ba > , numbers )ba(A 004 = (112) )ba(A 20
202 = (113)
are primsnumbers
Primitive Pythagorean triads ,which halt a value primes numbers by a rule C
R U L E
If
2
2
)7(R7
=
(114)
divide numbers
)ab( 202
0 (115)
-
43
and
)7(
R7
= (116) divide numbers
)ab( 00 (117) then are
05 cA = (118) primesnumbers
Explanatory or demonstration examples for calculations of prime-numbers useful formulas:
05
00004
00003
002
02
02
002
02
01
cAbaif)ba(Aabif)ab(A
baif)ba(A
abif)ab(A
=>=>=>=>=
(119)
Lets take from the book [1], see page17, ready triads of primitive Pythagorean triads
:)c,a,b( 000
)61,11,60(:3)29,21,20(:2
)5,3,4(:1
)65,33,56(:6)37,35,12(:5
)13,5,12(:4
)65,63,16(:9)41,9,40(:8)17,15,8(:7
)73,55,48(:12)53,45,28(:11
)25,7,24(:10 (120-33)
Substitute these numbers into my formulas and you will get
a full set of prime-numbers, born by corresponding of following rules:
Primitive Pythagorean triads, which
halt a value primes numbers by a rule A
Triad 2 :
41400441)ba( 202
0 == prime number 12021)ba( 00 == prime number
29cA 05 == prime number
-
44
Triad 4 :
177/11925144)ab( 202
0 === prime number 134)ab( 00 == prime number
13cA 05 == prime number
Triad 5 :
10811441225)ba( 202
0 == prime number 231235)ba( 00 == prime number
37cA 05 == prime number
Triad 7 :
237/16164225)ba( 202
0 === prime number 7815)ba( 00 == prime number
17cA 05 == prime number
Triad 8 :
2177/1519811600)ab( 202
0 === prime number 31940)ab( 00 == prime number
41cA 05 == prime number
Triad 12 :
1037/72123043025)ba( 202
0 === prime number 74855)ba( 00 == prime number
73cA 05 == prime number
Primitive Pythagorean triads, which halt a value primes numbers by a rule B
-
45
Triad 6 :
135/65)5/( 0 == prime number 204710893136)ab( 20
20 == prime number
233356)ab( 00 == prime number
Triad 9 :
135/655/c0 == prime number 37132563969)ba( 20
20 == prime number
471663)ba( 00 == prime number
Triad 10 :
55/255/c0 == prime number 52749576)ab( 20
20 == prime number
17724)ab( 00 == prime number
Primitive Pythagorean triplets, which halt a primes numbers by a Rule
Triads 3 :
If 2
2
)7(R7
=
divide number
7149/34791213600)ab( 202
0 === prime number and
)7(R7
=
divide number
-
46
77/491160)ab( 00 === prime number then
61cA 05 == Prime number
The remainder example
Triad 13 : If v=17 and u=14 , then according (13):
485c476b93a
0
0
0
===
9272176898576226)ab( 202
0 == prime number 383)93476()ab( 00 == prime number
975/c0 = prime number It is Rule B
Triad 14 :
If v=17 and u=12 , then according (13):
433c408b145a
0
0
0
===
777207/43914502521464166)ab( 202
0 === prime number 263145408)ab( 00 == prime number
433c5A 0 == prime number It is Rule A
Triad 15 :
If v=18 and u=11 , then according (13):
445c396b203a
0
0
0
===
60711520941816156)ab( 202
0 == prime number 193203396)ab( 00 == prime number
895/cA 05 == prim number It is Rule B
Triad 16 :
If v=18 and u=13 , then according (13):
493c468b155a
0
0
0
===
857277/99919402524024219)ab( 202
0 === prime number
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47
313155468)ab( 00 == prime number 493cA 05 == prime number
It is Rule A
Et cetera, et cetera .
P A R T 5
Intercommunication
a between the elliptical curves, Abel group
and the non-modular forms.
According [1], chapter 11, section A , the key moment is to connect information of elliptical curve of Frey with analytical function of complex variable and global invariant infinite
product of simple numbers p :
As realization that statement I propose a solution possible variant of equation Freys-Yaroshs:
)cba(
)BX(X)AX(Yn
on
on
o
2
==+=
where
n
0
no
n0
c)BX(
bX
a)AX(
=+=
=
And
22o
o
22o
uvcvu2b
uva
+==
=
primitive Pythagorean triplets.
In this case we have endless series non-modular curves.
Because:
Here every number from endless series prime numbers
)ab(A 202
01 = if oo ab > will not divide into
222
22
0
0
)()2(
)122b(4)vu2b(416
=======
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48
If number
2
n4ooo )cba(16
=
is function a discriminant my elliptic curves:
8
n2ooo
2)cba( =
at the same time )ab(A 202
01 = will not divide
3
22o
220
3)12a(9
)uva(927
====
===
if odd number 27 is equivalent:
+= 2
o
3o
ba427
where )b27a4( 2o
3o +=
discriminant for canonical form any elliptic curves:
oo32 bxaxY ++=
With all this going on to make good use multitude
construction to analytical function of complex variable:
20
20
20
0000
ba
)bia()bia(ssS
=+==+==
and
nn0
n0
n0
n0
n0
n0nnn
zba
)bia()bia(ssS
=+==+==
as general common invariants for systems co-ordinates A and B It is complex functions.
I propose three of Rule for computation
all simple (prime) numbers : Rule A, Rule B and Rule C.
See Part 2
And following research result a Riemanns statements.
Riemanns statement 1
Function:
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49
1)s1()s1(2sSin2)s( 1ss =
determines all complex numbers 1s
My statement 1
Function:
1)s1()s1(2sSin2)s( 1ss =
determines all complex numbers 1s if
1ba
)bia()bia(ssS2
02
02
0
0000
=+==+==
See Pic.2
where:
2o
2o
2o
22o
o
22o
cba
uvcvu2b
uva
=++=
==
For all pairs number uv > of various evenness taken from endless series of natural
numbers N= 1, 2, 3, 4, 5,..
Riemanns statement 2
)s(Z function determines the seros for ...,6,4,2s =
My statement 2
0SQSin)S(Z == function determines endless series zeros 0bo = for
...,S6,S4,S2S =
If angle Q=0 , see Pic.2 and Pic. A , then:
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50
0bi0bi
o
o
==+
1a
)bia()bia(ssS2
0
0000
===+==
what
1cba
1uvc0vu2b
1uva
2o
2o
2o
22o
o
22o
==+=+=
====
where v=1 and u=0 taken from:
==
,...5,4,3,2,1,0u...,5,4,3,2,1v
Geometrical interpretation, see Pic. A
Pic. A
Riemanns statement 3
Forma:
=
= 1n sn)n(
)s(1
determines stripe
1)s(R0
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51
for all untrivial zeros, when
Rt,ti21 +
My statement 3
Let 2S2 = where:
++++++= nif2...2...2222 n32102 then:
+++++=+= nif1....2....222.1 n321222 where:
r21
2 += and
++++= nif5.0....2...22r n32 Forma:
=
= 1n sn)n(
)s(1
determines stripe 1)s(R0
for all untrivial zeros, when
Rr,ri21 +
as equivalent:
R,i21 +
See Pic. B
Pic. B
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52
Here:
==
==
==+
+=+=
+=+=
+=+=
............
5.1i21)i(z3intPo
1i21)i(z2intPo
5.0i21)i(z1intPo
...........
5.1i21)i(z3intPo
1i21)i(z2intPo
5.0i21)i(z1intPo
R e s u l t
Forma:
=
= 1n sn)n(
)s(1
determines stripe
1)z(R01)z(R0
for all untrivial zeros, see Pic.2 and Pic. A, when, angle 0aQ o == : 0)S1()S1(
2SSin2)S( 1SS ==
and complex function :
0b)bia()bia(ssS 2o0000 ==+== because:
0vu2bo == where v=1 and u=0 taken from:
==
,...5,4,3,2,1,0u...,5,4,3,2,1v
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53
Consequently, exist endless series complex numbers
as endless series of zeros:
0)bia(iz0)bia(iz
oo
oo
===++=
Forma:
0n
)n()S(
11n
S=
=
contain function: +++++== nif1...2...222)n( n3212
and complex function:
1a
)bia()bia(ssS2
0
0000
===+==
what
0vu2b1uva
o
22o
====
where v=1 and u=0 taken from:
==
,...5,4,3,2,1,0u...,5,4,3,2,1v
Consequently, determines zeta function:
==
1n
S0
n)n(
1)S(
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54
P A R T 6
Riemanns sphere as mapping of space task common solutions of Conjecture
Birch and Swinnerton-Dyer and Riemanns Hypothesis
1. Riemanns sphere
Lets address to Pic.3 and Pic.4.
If complex number oo bias += , see Pic. 2, is set on a plane ),( by a point )b,a(P oo the point )Y,,('P of crossing of a piece PN
of a straight line with a surface of Riemanns sphere is a new geometrical representation of number oo bias += in the Decartes system
of coordinates )Y,( . All complex numbers oo bias += and, see Pic.2, Pic.4, Pic.5, are displayed by points which projections
to a plane of complex numbers += iz and = i'z oo bias = lay inside of a semicircle of diameter, see Pic.3a.
Here
1)zRe(0
Pic.3a
Remark
Authors of Birch and Swinnerton-Dyer conjecture approve:
if (1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if (1) is not equal to 0,
then there is only a finite number of such points.
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55
2. General mathematical strategics a solution of the Birch and Swinnrton-Dyer Conjecture
and Riemanns Hypothesis
Defining role in the proof by Birch and Swinnerton-Dyer belongs to the EXPANDED or CLOSED plane of complex numbers:
+= iz and = iz properties of this plane are defined by properties of Riemanns Sphere
below, on Pic.4 Riemanns sphere [13] is represented.
Pic. 4
Here : Flatness ),( for complex numbers += iz and = i'z ;
Point )z,z(PP = ; Point )Y,,('P'P = ; Axis absciss for ; Axis ordinat for ;
It is stereographycal projection flatness ),( to sphere )Y,,(
Stereographycal projection keep angles . Thats have mathematical description, see (32):
2o
2o
2o
2o
2o
2o
2o
o2
o2
o
o
20
02
o2
0
o
c1c
ba1bay
c1b
ba1b
c1a
ba1a
+=+++=
+=++=+=++=
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56
In a Fig. 5 it is presented diametrical section of Riemans sphere
Pic.5
3. INITIAL DATA
V A R I A N T D A T A 1
If angle =Q , ( see Pic.2 , Pic.3, and Pic.6) , then:
0)s(Re0)s(Re
ibib
o
o
==++
(9)
According we have endless series numbers:
2o
oooo
ibaibassS
==+==
(10)
as endless series primitive numbers of Pythagora over field N natural numbers )uv( > :
)uv(S 22o +== (11) In the end we have endless products,
as endless series seros:
00sinsss,sS === (12) According this statement, we create endless series
functions
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57
for proof a Birch and Swinnerton-Dyer conjecture
0)S1()S1(2Ssin2)S( 1SS == (13)
Here scalars 0S = are arguments for theyre functions .
Pic. 6
The axis of absciss oa is geometrical axis symmetry for two a system coordinat.see Pic. 6.
Common the angle Q is general argument for general functions: 0)S1()S1(
2Ssin2)S( 1SS == (14)
V A R I A N T D A T A 2
If angle 0Q = , see Pic. 7, then:
o
o
o
o
a)s(Rea)s(Re
0ib0ib
==
==+
(15)
According we have endless series numbers:
20
0000
a
)bia()bia(ssS
==+==
(16)
as endless series primitive numbers of Pythagora over field N natural numbers:
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58
)uv(Sa 22o == (17) In the end we have endless vectorials products, as endless series sero:
00sinss]s,s[S === (18) According this statement, we create endless series functions
for proof a Birch and Swinnerton-Dyer conjecture:
0)S1()S1(2Ssin2)S( 1SS == (19)
Here scalars 0S = are arguments for theyre equations. 0bi+
oib Pic. 7
Birch and Swinnerton-Dyer Conjecture
Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like
x2 + y2 = z2
Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function (s) near the point s=1. In particular this amazing conjecture asserts that if (1) is equal to 0, then there are an infinite number of rational points (solutions), and
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59
conversely, if (1) is not equal to 0, then there is only a finite number of such points.
4. SPECIAL PROPERTIES OF COMPLEX NUMBERS
The member of Russian Academy of Sciences, famous mathematician G. Pontryagin, see. [10] and Pic.2 Pic.7,
learning properties of complex numbers :
=+=
=+=
i'ziz
orbias
bias
oo
oo
(21)
discovered their polisemy, which was seen by G.V.Leibnitz in his time , as unexplained wonder.
Complex numbers can be in the same time: ) complex numbers,
) points representing these numbers on complex plane, ) vectors, corresponding to these numbers.
The length of such vectors is determined by module:
22
2o
2o
'zz
or
bass
+==
+==
(21)
Because of that there can be formulas given above.
Thus we have a possibility to consider primitive Pithagoras numbers 22
o uvc += as a basis of two forms of general complex invariants
(two forms of complex functions)
2
o2
o2
o2
o2
o c])bi(a([)ba(S ==+= (22)
220
20
0000
ba
)bia()bia(ssS
=+==+==
And also - as a basis of spectral invariant
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60
++++++= nif...)c...ccc( no3o2o10oo (23) At that whole numbers
22o uvSc +== make a basis
for infinite numbers of whole-numbered decisions of the system of equation:
zyx
nnn
n22n
CBA2nifcba
,...3,2,1,0nif)uv()uv(f
=+=+=+=>
(24)
5. New type solutions to equations for Birch and Swinnerton-Dyer Conjecture
+==>
++++==>+++==>++==>
+==>==>
nif)uv(c)uv(f
.......................................................................................vu4uv4uv6uvc)uv(f
vu3uv3uvc)uv(f
uuv2vc)uv(f
uvc)uv(f1c)uv(f
n22non
464644884o4
2424663o3
42242o2
2211
00
(19)
6. General proof
Birch and Swinnerton Dyer Conjecture
If angle =Q , see Pic.2 and Pic.3, then:
0)s(Re0)s(Re
ibib
o
o
==++
(20)
and also to data of 3 7, on the EXPANDED complex plane of numbers: += iz and = i'z (21)
all real coordinates 0= . Hence, in the Decartes system of coordinates )Y,,( is formed a line
of infinite set of ZERO: 0)zRe( = and 0)'zRe( = (22)
Similar picture we have in the system of coordinates represented on Pic. 2: 0)sRe( = and 0)sRe( = (23
thus, in a point 1= , see Pic.5, function is formed:
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61
0)1()1(2
sin2)( 1 == (24) on the basis of spectral invariant:
+=
=++++++==++++++=
nif1...)c...cc(1
...)c...ccc(n
o3
o2
o1
o
no
3o
2o
10
oo
(25)
From which definitions of ordinate follow:
)(1 == (26) and corresponding functions:
0)11()11(2
1sin2)1( 111 == (27) As a result we receive return display from the EXPANDED complex
plane ),( on a plane of complex numbers:
oo
oo
biasbias
=+=
(28)
systems of coordinates, represented on Pic.2. It means, that spectral :
+=
=++++++==++++++=
nif1...)c...cc(1
...)c...ccc(n
o3
o2
o1
o
no
3o
2o
10
oo
(29)
Is put inconformity ton infinite number of primitive Pythagoras numbers :
++++++
nif...)c...ccc( no
3o
2o
10
oo
(30)
and corresponding infinite row of integer decisions of the equations of the First type, see (19).
Remark.
Spectral invariant is formed over field N of natural numbers. It does not depend on a choice of coordinates and on operation
of display of complex numbers on the EXPANDED plane of complex numbers.
If
oo
oo
biasbias
=+=
(31)
then we have stereographycal projection
flatness ),( to sphere )Y,,( and keep angles, see Pic.4 .
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62
Thats have mathematical description
1c1
cc1y
c11
c1c
c1
2o
2o
2o
2o
2o
2o
2o
+=+=
==
=
(32)
if
triads of Pythagorean:
22
0
0
220
uvcvu2b
uva
+==
= (33)
for all paar v>u numbers are the numbers of various evenness taken from endless series :
,...4,3,2,1)1u(v,....,4,3,2,1,0u
=+==
(34)
and , see (4) :
2
oo2
o
2o
o2o
2
ucu2
buav
vcv2
bavu
==+=
=== (35)
In the end, also:
If 2
ocn= then we create relativity system coordinat: For operation of display of my complex numbers, see (28), (31) and Pic.2,
on the EXPANDED plane of complex numbers:
==+=+=i'zbia'sizbias
oo
oo (37)
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63
I offer two easy Rule:
==
+=+=
in
1nin1'z
in
1nin1z
n
n
(38)
for one in two endless series numbers:
;...n1ni
n1z;...,
32i
31z;
21i
21z;1z n321
+=+=+== (39)
;...n1ni
n1'z;...,
32i
31'z;
21i
21'z;1'z n321
==== (40) Endless series numbers (39) is enless series points:
n1
n = and n1n
n= (41)
Theyre lie to segment, see Pic.9: 1=+ (42)
Also if : n (43)
then: 0lim n = and 0lim n = (44)
and ii10)i(limzlim nnn =+=+= (45)
Concequence:
modulus 0izn = (46)
All theyre have geometrical interpretation,see Pic.8 :
Here 1)zRe(0 for +=+= ii)z(Rez
Pic.8
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64
Here striple 1)zRe(0
Pic.9
C o r o l l a r y
B.Riemann (1859) :
(s) function determines all complex numbers 1S . Correspondence:
4. Special properties of complex numbers
(s) function determines the seros for ,,,,6,4,2S = Functional equation
(47)
and forma
(48) determines stripe