PROOF for Beal Conjecture mainly by Simple Geometry

PART [1] LEMMA-(1) & PROOF for Lemma(1)

..... .....

[1A] LEMMA-1 states that for Beal conjecture C^z = A^x + B^y,

value of C & either one of A & B must be integer > 1 which implies that C^z is a Composite Number & either ONE of  A^x & B^y must be Composite number

.... .....

Proof for LEMMA-(1) is given below.

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By conditions of Beal Conjecture 

A & B are positive integers, which implies that  A^x + B^y > 1 

Also for C^z value of z is integer > 2 &

for C =1 value of C^z = 1

Therefore

A^x + B^y = C^z  and A^x + B^y > 1 implies that C must be integer > 1 

which implies that as claimed by Lemma-1 for Beal Conjecture  C must be an integer  more than 1 & C^z is a Composite number 

..... ..... .....

Also for Beal Conjecture term C^z value of  z is an integer > 2. which implies that

for valid term of C^z  minimum value of  z = 3 where C > 1 such that 

Minimum value of valid term for C^z = 2^3 = 8 &

value of C^z is integer 2^3 or more

...... ..... 

For A = 1 & B =1 value of  A^x + B^y = 1 + 1 =  2

therefore  A^x + B^y = integer 2^3 or more

implies that for Beal Conjecture both A & B can't be simultaneously = 1 

such that  as LEMMA-1 stated for Beal Conjecture C > 1 & Either ONE among A & B must be INTEGER > 1 

which implies that for Beal Conjecture C^z = A^x + B^y

along with C^z which is a Composite number, Either ONE among  A^x & B^y must be a COMPOSITE Number &

Lemma(1) is proved

.... ..... ....

For this article based on Lemma-1 we take along with C value of B > 1 and thus

for this article C^z & B^y are taken as Composite Number such that

PROOF for Beal Conjecture is obtained mainly based on PROOF for COMMON FACTOR in C^z & B^y

Thus in this article based on simple & basic rules in Geometry,

firstly PROOF for COMMON FACTOR in C^z & B^y is obtained

which directly prove COMMON FACTOR in C^z, B^y & A^x as given in [1B] below &

Based on PROOF for Common Factor in C^z, A^x & B^y this article have PROOF for Beal Conjecture

.... .... ..... ..... 

[1B] For  C^z = A^x + B^y

PROOF for COMMON FACTOR in C^z & B^y directly PROVE COMMON FACTOR in C^z, B^y & A^x as given below

Common factor in C^z & B^y implies that

C^z = K2*D & B^y = K1*D where K2 & K1 are INRGERS > 0 & D is an INTGER > 1 where D is COMMON FACTOR in C^z & B^y 

which implies that K2 > K1 because C^z > B^y

Therefore

A^x = C^z - B^y = K2*D - K1*D  = (K2-K1)*D where (K2-K1) is an INTEGER

which implies that 

If C^z & B^y have proof for COMMON FACTOR,

Then it also PROVES C^z, B^y & A^x have COMMON FACTOR

... ..... .....  ..  

PART [2] SKELETAL EXPRESSIONS of integers & its implications

.... ...... ...

[2A] SKELETAL EXPRESSIONS of INTEGERS & its based RECTANGLE or square FIGURE for every Natural Number

..... ....

In Series of Natural Numbers 1 is added to every integer to get the next INTEGER in that SERIES

which implies that ONE can be taken as BASIC UNIT of ALL natural Numbers such that every NATURAL Number T can be represented by a basic SKELETAL EXPRESSION as

T = 1 + 1 + 1 + ...... ..... .... up to T number of UNIT Terms & T = U*N

where U = value of a UNIT Term & N = Number of Unit Terms

Thus we have equation T = 1*T where 1 = value of Basic unit of Natural numbers such that RHS part T represents Number of Unit of Unit Terms

..... ........

[2B] For COMPOSITE NUMBERS, its MODIFIED SKELETAL EXPRESSIONS where value of UNIT Term U is INTEGER > 1

..... ..... .....

For a COMPOSITE number T, Equation T/T = 1 

implies that T is divided to T number of divisions & each division [Unit] have value = 1

which gives  equation T = 1*T & a BASIC SKELETAL EXPRESSION for T that have value of  Unit Term U =1 & Number of Unit Terms N = T

such that T = U*N = 1*T

If the same INTEGER T is divisible by N where N is an INTEGER < T, then

For T/N = U  value of U is an INTEGER > 1  such that N & U are 2 FACTORS in C^z 

such that

T/N = U implies that T is divided to N number of identical divisions [Units] of integer value U where U > 1

Thus for every composite number T

For N < T Equation T/N = U gives MODIFIED SKELETAL EXPRESSION for T as

T = U + U + U + ..... ...... ... up to N number of Unit Terms where U in an INTEGER > 1 &

T = U*N where U = value of Unit term & N = Number of Unit terms

which implies that for MODIFIED SKELETAL EXPRESSIONS for a COMPOSITE Number T

T = U*N = F1*F2 where U = F1 & N = F2 are INTEGERS >1 such that 

U= F1 & N = F2 are 2 Factors in T

..... ..... ... ..... 

[2C] SKELETAL EXPRESSION implied RECTANGLE/square figures for NATURAL numbers 

For every integer T equation T/N = U give T = U*N &

equation T = U*N gives RECTANGLE/Square figure for that integer such that

In the case of a composite number T,

there are one or more values for N < T and T is divisible by N < T such that 

For equation T/N = U value of  U > 1 such that T = U*N = F1*F2 where F1 & F2 are 2 Factors in T

that give a RECTANGLE or SQUARE Figure for T that have both SIDES to represent INTEGER > 1 &

For N = T  equation T/N = U gives value of U = 1 & T = 1*T which implies that

Every Natural Number have a General Rectangle 

that have breadth = 1 & Length = T itself for the equation T = U*N =1*T

..... ..... ....  .. .. 

PART [3] By keeping CONDITIONS of BEAL CONJECTURE

SKELETAL EXPRESSION & 

FIGURE by geometry for C^z in Beal Conjecture C^z = A^x + B^y

..... .......

For Beal Conjecture, z is an integer > 2 & by LEMMA-1 for a valid term C^z  value of C is an integer > 1 which implies that minimum term for C^z = 2^3 &

In the case of z which is an integer > 2

z = 2 + n where n is an integer > 0 & n = z-2

which implies that  C^z =  C^(2+n) = C^2*C^n  = C^2*C^(z-2)

Therefore in the case of  Beal Conjecture term  C^z, for all values of C & z, 

C^2 can be taken as conditions of Beal Conjecture implied Value of Unit term 

such that

equation C^z = U*N = C^2*(C^(z-2) give Beal Conjecture implied SKELETAL EXPRESSION for C^z = U*N = C^2*C^(z-2) that form as

C^z = C^2 + C^2 + C^2 + ....... ...... .... up to C^(z-2) Number of Unit Terms 

where value of unit U = C^2 &  C > 1 ..... ...[Proved by Lemma-1]

which implies that in Geometry

by conditions of Beal Conjecture C^z is a RECTANGLE that have its both SIDES to represent INTGERS > 1 & formed for C^z = U*N = F1*F2 where value of Unit term U = C^2 & Number of Unit Terms N = C^(z-2) are 2 factors in C^z such that

AREA of RECTANGLE for Beal conjecture term C^z can be divided to C^(z-2) number of Unit SQUARES of Side = C & AREA = C^2

which further implies that by keeping conditions of Beal Conjecture, for C^z

there isn't a General Rectangle formed for C^z = U*N =1*C^z  that have breadth represents integer 1 for U & Length represents C^z for N &

its related SKELETAL EXPRESSION for C^z = 1*C^z that have U = 1,& N = C^z which form as important difference between NATURAL Numbers and Beal Conjecture terms C^z as well as B^y

which particularity is used in this article to PROVE COMMON FACTOR in C^z & B^y 

that is used in this article to get PROOF for BEAL Conjecture

..... ...... ..... ...

[3A] By keeping Conditions of Beal Conjecture,

for SKELETAL EXPRESSION of C^z where C^z = U*N

Minimum value of Number of Unit Terms = N & 

Minimum value of Unit Term = U

..... ......

By Part [1] & Part [3]

For valid Beal Conjecture term C^z  value of C > 1 & z > 2 

which implies that

Minimum value of C =  2, Minimum value of z = 3 &

Minimum Value of C^z = 2^3 which implies that 

Also for minimum value C = 2 & z = 3

C^z = U*N = (C^2)*(2^(z-2) = (2^2)*[2^(3-2)] where N = [2^(3-2)] = 2 & U = 2^2

Therefore

By keeping conditions of Beal Conjecture for C^z = U*N

Minimum value of Unit term U = 2^2 & 

Minimum value of Number of Unit terms N = 2 

such that for Beal Conjecture term C^z = U*N

value of U is an integer 2^2 or more &

value of N is an integer 2 or more

..... .......

which implies that for  C^z = U*N 

equation C^z = 1*C^z where U =1implies value of N = C^z as well as

equation C^z = C^z*1 where N = 1 implies value of U = C^z  

such that by keeping conditions of Beal Conjecture U = 1 & N = 1 are INVALID

which implies that by keeping conditions of Beal Conjecture

values of  U =1 & U = C^z are INVALID as well as

value of N =1 & N = C^z are INVALID such that

for Beal Conjecture term C^z = U*N values of  U & N are integers more than 1 & less than C^z and C^z = U*N = F1*F2 where F1 & F2 are 2 Factors in C^z

Which implies that

Equation C^z = U*N = 1*C^z where U =1 & N = C^z as well as

C^z = U*N = C^z*1 where U = C^z & N = 1 are IRRELEVANT & insufficient equations to prove RULES/particularities related to Beal Conjecture terms C^z 

like verification of COMMON FACTOR in C^z & B^y

..... ....... ......  ...   

[3B]  By keeping Conditions of Beal Conjecture,  SKELETAL EXPRESION & RECTANGLE figure for B^y

..... ..... ....

By LEMMA-1 along with C & C^z we have taken B > 1 & B^y as COMPOSITE NUMBER 

which implies that C^z & B^y are similar terms where C & B are integers > 1 & its exponents x & y are integers > 2 

which implies that as in the CASE of  Beal conjecture term C^z also for B^y 

Conditions of Beal Conjecture implies a SKELETAL EEXPRESSION such that

B^y = B^2 + B^2 + B^2 + ..... ..... .... up to B^(y-2) NUMBER of Unit terms &

its implied RECTANGLE that have both SIDES to represent INTEGERS > 1 for the Equation B^y = U*N = B^2*B^(y-2)

which implies that

by keeping conditions of Beal Conjecture

C^z as well as B^y haven't a Rectangle and Skeletal Expression that have value of Unit term U =1 where C^z = U*N =1*C^z and B^y = 1*B^y such that

By keeping conditions of Beal Conjecture Rectangle for C^z and B^y must have Length and breadth to represent integers >1 

.... ...... ....  ....   

PART [4A] BASIC & simple RUELS in GEOMETRY/MATHS that are used in this article to PROVE COMMON FACTOR in Beal Conjecture Terms C^z & B^y

where C^z & B^y are COMPOSITE Numbers......  (verified by Lemma-1)

... ..... ...

By RULES in geometry

if COMPOSTE Numbers C^z & B^y can be represented by Rectangles/square Figures that have it's SIDES to represent values in INTEGERS, 

then EQUAL SIDES that represent INTEGER > 1 of such ONE each RECTANGLES formed for C^z & B^y

VERIFY COMMON FACTOR in C^z & B^y 

Also such verification for Common factor by one each Rectangle for C^z & B^y is VALID for all cases of other Rectangle/square figures [for C^z & B^y ] and sufficient to PROVE common factor in C^z & B^y

.... ..... ...

which further implies that by the said RULE to VERIFY COMMON FACTOR in C^z & B^y,

Rectangle formed for Bigger term C^z for the equation C^z = 1*C^z is insufficient to verify COMMON FACTOR in C^z & B^y

because all FACTORS in B^y are less than C^z & for the rectangle formed for C^z = 1*C^z, 

NO FACTOR in C^z that is less than C^z is projected for verification of common factor in C^z & B^y where

Equation C^z = U*N = F1* F2 can project every FACTOR in C^z as U = F1 or N = F2 to complete VERIFICATION of COMMON FACTOR in C^z & B^y

...... .....  .... 

Thus as explained in [3A] for Beal Conjecture term C^z RECTANGLE for C^z = 1*C^z is IRRELEVANT to verify RULES related to Beal Conjecture 

as well as to VERIFY COMMON FACTOR in C^z & B^y by methods in Geometry as explained above in this part [4]

Ex: T1 = 15 & T2 = 18 where 18 = 15 + 3

Rectangle for 18 = 1*18 is INSUFICIENT to verify Common Factor in 18 and 15 because all Factors in 15 are less that the projected factor 18 in 18 

Among other equations for such that 18 = 2*9 and 18 = 3*6

Rectangle formed for 18 = 3*6 & 15 = 3*5 VERIFY and PROVE COMMON FACTOR 3 in 18 & 15 by equal SIDES for 3

[Also in Arithmetic when 18 & 15 are expressed as multiple of factors such that equal factors verify common factor in 18 & 15 bigger number 18 expressed as 18 = 1*18 is insufficient expression to project possible equal Factors in 18 & 15 to verify common factor in 18 & 15]

...... ...... ........` 

PART [4B] RULES in Maths/Geometry, To verify COMMON FACTOR in T2 & T1

where T1 & T2 are 2 COMPOSITE NUMBERS such that T2 > T1 &

Equation/Rectangle for T2 is given for T2 = 1*T2

.... ... .... 

By Rules in Maths 

If bigger number T2 is given by a RECTANGLE for T2 = 1*T2 that Rectangle must be MODIFIED as Rectangles for T2 = F1*F2 such that F1 & F2 are 2 FACTORS in T2

to complete verification of COMMON FACTOR in T2 & T1

which implies that

In the case of Beal Conjecture C^z which is a Composite number,

Equation C^z = 1*C^z is IRRELAVANT to verify COMMON FACTOR in C^z & B^y

If Beal Conjecture term C^z is given as C^z = 1*C^z,

equation C^z = 1*C^z must be MODIFIED as C^z = F1*F2  to verify COMMON FACTOR with B^y

where equation C^z = F1*F2 can project every FACTOR in C^z as F1 or F2 &

thus give ALL valid Rectangles of C^z to complete verification of  COMMON FACTOR in C^z & B^y 

Therefore based on the RULE as said above 

to verify & PROVE COMMON FACTOR in C^z & B^y by method as said above  

Rectangle/Skeletal Expression formed for C^z for the equation C^z = I*C^z is IRRELEVANT/insufficient &

Rectangles/Skeletal Expressions formed for C^z for the equation C^z = F1*F2 are SUFFICIENT 

to verify whether there is COMMON FACTOR in C^z & B^y

..... ...... ....  

PART [5] This Article have PROOF for COMMON FACTOR in Beal Conjecture terms C^z & B^y based on 

(1) RECTANGLE figure of  C^z for C^z = F1*F2  where length of Sides represent F1 & F2 

(2) STRUCTURAL Figure for C^z based on SKELETAL expression

for C^z = U*N = C^2*C^(z-2) where Unit term U = C^2 is represented by area of Unit Squares & 

its modified equations for C^z = U*N = F1*F2 where U = F1 & N = F2 are 2 Factors in C^z such that Number of Unit terms = F2 must be integer > 1 &

Value of Unit Term must be integer > 1 where

U = F1 is represented as Unit Rectangle such that F1 = Area of Unit Rectangle,

as explained below in Part [5A] onward

..... ...... ..... ...  ...  

[5A] SIMPLE Concept that is used in this article to get PROOF for COMMON FACTOR in C^z, A^x & B^y,

based on which this Article have a PROOF for Beal Conjecture

..... ..... .....

Let there are 7 identical Boxes that have 3 Oranges and 2 Apples in Each box, 

2, 3 & 7 have NO COMMON FACTOR,

But Number of identical boxes 7 in the system as above cause COMMON FACROR in Total number of Oranges [T1], Total Number of Apples [T2] & Total number of FRUITS [T3] in the system such that

T3 = T1 + T2 where T1 = 7*3 T2 = 7*2 & T3 = 7*(3+2)

..... ........

Similarly as explained in [3A]

For Beal Conjecture term C^z = C^2 + C^2 + C^2 + ...... .... ... up to C^(z-2) number of Unit Terms where C^z = U*N = C^2*C^(z-2),

by Geometry U = C^2 can be represented as IDENTICAL SQUARES where Number of Unit Squares N = C^(z-2) &

for VERIFICATION of COMMON FACTOR in C^z & B^y

equation C^z = U*N = F1*F2 includes U = F1 = C^2 & N = F2 = C^(z-2) 

which implies that

To VERIFY COMMON FACTOR in C^z & B^y

RECTANGLE for C^z = U*N = C^2*C^(z-2) need to be MODIFIED for C^z = U*N = F1*F2 where value of Unit U = F1 and number of Unit Terms  N = F2 are 2 FACTORS in C^z such that

every FACTOR in C^z can be projected as U = F1 for completion of VERIFICATION as COMMON FACTOR in C^z & B^y

such that 

by geometry U = F1 can be represented as Unit Rectangles where Area  of Unit Rectangle = F1 which implies that there are F2 number of  IDENTICAL Unit Rectangles of Area = F1

Where C^z = U*N = F1*F2 implies a SKELETAL EXPRESSION for C^z

as C^z = U*N = F1 + F1 + F1 + ..... ...... ...... up to F2 number of Unit Terms where

U = F1 include factor = C^2 & 

number of Unit Rectangles  N = F2 includes factor = C^(z-2) 

..... ....... .....  ..    

[5B] Based on SKELETAL Expression for C^z where C^z = U*N = F1*F2

STRUCTURAL FIGURE for C^z = U*N = F1*F2 where C^z = A^x + B^y  and

VERIFICATION of COMMON FACTOR in C^z & B^y

....... ....... .........

For the Skeletal Expression for C^z = U*N = F1*F2 where

C^z = F1 + F1 + F1 + ...... ..... ..... up to F2 number of Unit Terms

By GEOMETRY unit term U = F1 (which also include F1 = C^2) can be represented as Area of a Unit RECTANGLE that have breadth =1 & Length = F1 where F1 = 1*F1 

which implies that there are F2 number of Unit RECTANGLES such that Area of Rectangle for C^z can be DIVIDED to Unit Rectangles where C^z = U*N = F1*F2

.... ....

By GEOMETRY all unit RECTANGLES are IDENTICAL &

Thus C^z = A^x + B^y  & C^z = U*N = F1*F2 implies that

A^x & B^y can be IDENTICALLY distributed in each Unit RECTANGLE of area = F1 

such that each Unit Term F1 is BIFURCATED identically as F1 = t1 + t2 where

t1 is PORTION of B^y per Unit RECTANGLE & 

t2 is PORTION of A^x per UNIT RECTANGLE

which implies that

BEAL CONJECTURE Term C^z have a STRUTURAL Figure SIMILAR to the system of 7 IDENTICAL BOXES that have ORANGES & APPLE placed IDENTICALLY as explained above in Part [5A] & 

where Number of IDENTICAL BOXES in the said system cause COMMON FACTOR in T3, T1 & T2 of the equation T3 = T1 + T2  related to that system

Similarly

For the STRUCTURAL Figure and SKELETAL EXPRESSION formed for Beal Conjecture Term C^z where C^z = U*N = F1*F2 

There are F2 number of identical UNIT RECTANGLES of area = F1 &

Unit RECTANGLES can be IDENTICALLY BIFURCATED as F1 = t1 + t2 such that

t1 is portion of B^y per UNIT RECTANGLE where t1 = B^y/F2 &

t2 is portion of A^x per UNIT RECTANGLE where t2 = A^x/F2 which implies that

B^y = t1*F2 & A^x = t2*F2 where C^z = F1*F2 such that 

Number of UNIT Rectangles = F2 cause COMMON FACTOR in C^z, A^x & B^y 

as explained below in part [7] onward

..... ...... .......   

[5C] EQUATION that give all RECTANGLES for C^z = U*N = F1*F2 &  

SKELETAL expressions/STRUCTURAL figures for C^z that are VALID for VERIFICATION of COMMON FACTOR in Beal Conjecture terms C^z & B^y

... ....  ....  

C^z/N where N is an INTEGER and C^z is DIVISIBLE by N implies that

C^z/N = U  &  C^z = U*N  where U is an INTEGER such that proper values of N

give ALL RECTANGLE for C^z that have its both sides to represent INTEGERS for the equation C^z = U*N 

which also implies a SKELETAL EXPRESSIONS for C^z = U*N as

C^z = U + U + U + ... ... .... up to N number of terms where value of U is an INTEGER 

which implies that

Based on value of N there are 2 CASES for the equation C^z/N = U & C^z = U*N 

such that  case (1) N = C^z & case(2)  N < C^z  as given below

.... .... ...... 

For CASE(1) where N = C^z value of U = 1 & C^z = 1*C^z

which is INSUFFICIENT equation to Very COMMON FACTOR in C^z & B^y by RECTANGLES ( as explained in Part [4] 

such that by rules in Mathematics

equation C^z = 1*C^z need to be modified to C^z = F1*F2 to VERIFY COMMON FACTOR in C^z & B^y ]

.... .... ...

Also for Beal Conjecture term C^z equation C^z = U*N = 1*C^z implies U = 1 and N = F2 which gives an INVALID SKELETRAL Expression for Beal Conjecture term C^z as explained in Part [3A]

such that Beal Conjecture implies a SKELETAL EXPRESSION for C^z = U*N

where U & N are INTEGERS > 1 which implies that

for Beal Conjecture Term C^z = U*N

U & N are integers MORE than 1 & LESS than C^z  where C^z = U*N = F1*F2

such that for U = C^z value of N = 1 

Similarly for N = C^z value of U = 1 where by keeping conditions of Beal Conjecture

U  & N must be more than 1 ....... ( as explained in part[3A]

..... .....

CASE(2) where N < C^z  implies value of U > 1 and  C^z = U*N = F1*F2

Where F1 & F2 are 2 factors in C^z

which implies that for C^z

CASES (2) give VALID RECTANGLES & SKELETAL EXPRESSIONS for C^z  to verify whether there is COMMON FACTOR in Beal Conjecture terms C^z & B^y

such that Rectangle for C^z have both sides to represent integers > 1 &

For Skeletal Expressions for C^z have number of Unit Terms as integer 2 or more &

value of Unit Term is integer > 1 such that every relevant factor in C^z can be projected 

as U = F1 

which implies that 

for N < C^z equation C^z/N = U & C^z = U*N = F1*F2 give

STRUCTURAL FIGURE for C^z as explained in [5B] that can be used to VERIFY Common Factor in C^z & B^y

where Unit term U = F1 is bifurcated as F1 = t1 + t2 such that there are F2 number of Unit term where N = F2 is an INTEGER 2 or more &

For C^z = U*N = F1*F2

B^y = t1*F2 & A^x = t2*F2

..... ...... ...  ... ...  ..... 

[5D] As verified in Part [1] for Beal Conjecture C >1 & C^z is Composite Number which implies that

a Composite Number T can be expressed as T = 1*T as well as T = F1*F2 where F1 & F2 are2 Factors in T

Therefore ,

Essentially Beal Conjecture term C^z have a SKELETAL EXPRESSION as explained in Part [5B]  for C^z = U*N = F1*F2 where U*N = F1*F2 = (t1 + t2)*F2

that can be used to verify Common Factor in C^z & B^y

..... ...... 

In this article we use also equation C^z/K = D which is identical equation to equation  C^z/N = U where C^z = D*K  & which give

Rectangle for C^z = D*K & SKELETAL Expression for C^z as

C^z = D + D + D + ...... .... .... up to K number of Unit terms where value of Unit Term  D is an INTEGER & 

for K < C^z equation C^z = D*K = F1*F2 gives Structural Figure for C^z

...... ....

Which implies that 

For equation C^z/K = D where C^z = D*K

equation C^z/C^z = 1 where N = C^z gives  D = 1 such that 

D = 1 & K = C^z give Rectangle for C^z = 1*C^z & 

Skeletal Expression where D  =1 & K = C^z as

C^z = 1 + 1 + 1 + ..... .... up to C^z number of terms 

which is IRRELEVAT to verify COMMON FACTOR in Beal Conjecture Terms C^z & B^y such that

Only For the value an INTEGER <  C^z  & C^z is DIVISIBLE by that INTEGER < C^z

We have EQUATIONS that give VALID RECTANGLES & SKELETAL EXPRESSIONS for Beal Conjecture term C^z  

to VERIFY COMMON FACTOR in C^z & B^y by methods & conditions as explained in Part[4] & Part[3]

...... ....  ....... .....  .....  

PART [6]:  LEMMA-3 for Beal Conjecture term C^z, and

PROOF for COMMON FACTOR in Beal Conjecture terms C^z & B^y

For LEMMA=3,  C^z is represented by a Rectangle that have both sides to represent integers >1 

Which implies C^z = F1*F2 where F1 & F2 are 2 factors in C^z 

 .... ...... ...

By Lemma-1, which is verified in PART[1]

for Beal Conjecture term C^z value of C is an integer > 1 

and for Beal conjecture term C^z value of z is an integers >2

which implies that for C^z = (C)*C^(z-1) 

essentially C^z have a Rectangle that have both SIDES to represent integers > 1 

Also as explained by Part [3] in this article,

By conditions of Beal Conjecture there isn't a Rectangle for C^z =  U*N = 1*C^z &

C^z forms for C^z = U*N = C^2*C^(z-2 such that 

for an integer N < C^z such that C^z is divisible by N

equation C^z/N = U implies value of U is an integer > 1 & C^z = U*N = F1*F2 where F1 & F2 are 2 factors in C^z

such that C^z = F1*F2 & give ONE or more Rectangles for C^z that have both SIDES to represent integers > 1  

Also as explained in Part [4] for VERIFICATION of COMMON FACTOR in C^z & B^y 

Rectangle formed for C^z = 1*C^z is IRRELEVANT such that

for the said verification

only Rectangles for C^z = F1*F2 are VALID to compare with Rectangles of  B^y

...... ...... .......

By keeping the conditions & Restrictions as said above, 

For LEMMA-3 Beal Conjecture term C^z is represented by a Rectangle that have its BOTH SIDES to represent INTGERS > 1 for C^z = F1*F2 and

for Lemma-3, Rectangle formed for C^z = 1*C^z is NOT allowed 

such that 

For LEMMA=3 equation C^z = T1 + T2 where C^z is a composite number,

T1 & T2 are integers such that also T2 is a COMPOSITE number

Therefore C^z = T1 + T2 implies that

Area of LEMMA-3 Rectangle for C^z can be bifurcated to 2  portions such that ONE portion represents T2 such that

the remaining portion represents T1 and

Lemma-3 states that for the LEMMA-3 Rectangle of a FIXED Value for  C^z 

where C^z = T1 + T2 

C^z,T2  & T1 have COMMON FACTOR

...... ...... .....

[6A] For Lemma-3 equation C^z = T1 + T2

To verify common factor in C^z & T2 all Rectangles of C^z formed for C^z = F1*F2 can be considered to compare with Rectangles formed for T2 that have both SIDES to represent integers such that

if ONE each of such Rectangles for C^z & T2 have verification for Common factor in C^z & T2

that verification is SUFFICIENT to PROVE COMMON FACTOR in C^z & T2

...... ..... ..... 

[6B] LEMMA-3 Equation C^z = T1 + T2

RELATION EQUATION of T2 with C^z and PROOF for LEMMA=3

..... ....... ....

For a positive integer N < C^z such that C^z is divisible by N 

Equation C^z/N = U where C^z = U*N implies that N & U are integers > 1 & give LEMMA-3 Rectangle for C^z

where C^z = T1 + T2 implies that

T2 can be represented as a portion of Area of Rectangle for C^z  & 

as a portion of C^z

T2 = L/K*C^z where L & K are 2 integers such that L < K  ( Because T2 < C^z) 

.... ....

For the term L/K,  meaningless equal factors in Numerator L & Denominator K can be avoided 

which implies that L & K are integers without COMMON PRIME FACTORS and

for the equation T2 = L/K*C^z  value of T2 is an integer 

which implies that to equate with integer value of T2

for equation T2 = L/K*C^z  its RHS part L/K*C^z must be an integer where L & K are integers without Common prime Factors

which implies that C^z must be divisible by K  and C^z/K = D  such that

T2 = L*D  where D is an integer 

..... ...... ........ 

Thus RELATION EQUATION of T2 with C^z forms as

T2 = L*D where D is an integer = C^z/K and C^z is divisible by integer K

..... ..... .....

Also D = C^z/K gives C^z = D*K  

which implies that for LEMMA-3 Equation C^z + T1 + T2

For the value of K that give integer value to term C^z/K such that C^z/K = D

There are RECTANGLES for C^z = D*K & for T2 = D*L such that 

Both of the rectangles for C^z = D*K & T2 = D*L  have SIDES that represent D as Equal SIDES 

which further implies that Rectangle for T2 can be represented as an INNER AREA Rectangle to Rectangle for C^z 

by sharing the EQUAL SIDES that represent D as COMMON SIDE 

where C^z = K*D & T2 = L*D

.... .......   ..... 

[6C] Based on Relation Equation  T2 = D*L where  C^z = D*K,

PROOF for COMMON FACTOR in C^z & T2

...... .....

C^z = D*K & B^y = D*L implies that C^z & T2 are multiples of integer D 

where C^z is a Composite number & D is an integer for D = C^z/K such that C^z is divisible by integer K

which implies that 

for Equations C^z/K = D gives C^z = D*K, & for different values of K

Equation C^z = D*K give ALL Rectangles for C^z that have both SIDES to represent integers such that

For C^z/K = D  & Rectangles for C^z = D*K

based on the value of K & its related value for D

there are 2 CASES value of D & Rectangles for C^z = D*K as given below

.... .....

(1) GENERAL CASE which is relevant to ALL Natural numbers

where K = 1 and K = C^z

For K =1 value of D = C^z such that C^z = K*D = 1*C^z &

for K = C^z value of D = 1 such that C^z = D*K = 1*C^z

Which implies that as explained in PART [4]

for the GENERAL CASE as said above

C^z have INSUFFICIENT Rectangles to Verify COMMON Factor in C^z & T2 

.... ..... ......

C^z is a COMPOSITE Number which implies that

Besides General CASE as said above

there is ESSENTIAL CASE where K < C^z 

that give ONE or MORE VALID Rectangles for C^z = F1*F2 such that F1 & F2 are 2 Factors in C^z  and

to continue verify COMMON FACTOR in C^z & T2 as explained below

..... .....

(2) ESSENTIAL CASE where K > 1 & K < C^z

..... ......

Ex: As a composite number 

for C^z where C > 1 & z > 2 the least term = 2^3 = 8 &

for 8 besides GENERAL CASE Rectangle where 8 = 1*8 

There is ESSENTIAL CASE Rectangles for 8 = 4*2 

Also by keeping  conditions of Beal Conjecture C^z have ONE or more ESSENTIAL CASE Rectangles for

C^z = U*N = C*C^(z-1) where U = C and 

as explained in PART[3] for C^z = U*N = C^2*C^(z-2) where U = C^2 that can be modified as C^z = U*N = F1*F2 where U = F1 and F2 are 2 Factors in C^z

which implies that

For equation C^z/K = D where  C^z = D*K,  value K > 1 & K < C^z gives D >1

such that

there must have ONE or more ESSENTIAL CASE Rectangles for C^z = D*K where D > 1 and

for ESSENTIAL CASE 

equation C^z = D*K gives Rectangles for C^z that have BOTH SIDES to represent integers > 1 such that based on RELATION Equation of  T2 with C^z,

equations C^z = K*D and T2 = L*D give Rectangles have EQUAL SIDES that represent D > 1 and VERIFY COMMON FACTOR in C^z & T2

Therefore

as said above in this PART

for C^z besides the GENERAL CASE Rectangle for C^z = D*K = 1*C^z where D =1

there is ESSENTAIL CASE Rectangle for C^z = D*K where D > 1 such that

Rectangle for C^z = D*K = F1*F2 and its related Rectangle for T2 = L*D have verification of Common Factor in C^z amd B^y by the EQUAL SIDES for D>1

which implies that 

as shown in the Example for 18 & 15 in PART[4A] and By Rule in Part [6A]

where GENERAL CASE Rectangle for 18 = 1*18 haven't verification of Common Factor in 18 & 15 such that

among the ESSENTIAL CASE equations for 18 such that 18 = 2*18 & 3*18 

Rectangle for 18 = 3*6 and Rectangle for 15 = 3*5 verify & PROVE COMMON FACTOR in 18 and 15 by EQUAL Sides for 3

Similarly for equation C^z/K = D where C^z = D*K and T2 = L*D

value of  K < C^z gives value of  D >1 such that

Composite number C^z gives ESSENTIAL CASE Rectangle for C^z = D*K = F1*F2

where D = F1 and K = F2 are 2 Factors in C^z and integers >1 

which implies that for Beal Conjecture term C^z 

there are ONE or more

ESSETIAL CASE Rectangles for C^z = D*K & its one portion  T2 = D*L where D >1 that verify & PROVE COMMON FACTOR in C^z and T2  

..... .....

Ex: For C^z/K = D where C^z = 9^4

GENERAL CASE value of K = C^z give D =1 and 

Rectangle for C^z = D*K =1*9^4 which is INSUFFICIENT to VERIFY Common Factor in C^z & T2

For ESSENTAL CASE value of K = 3 give D = 3^7 where Rectangle for C^z = D*K and T2 = D*L form  for 9^4 = (3^7)*3 and T2 = (3^7)*L where L is an integers

which implies that for Lemma-3 equation C^z = T1 + T2

ESSENTCIAL CASE Rectangles 

VERIFY & PROVE Common Factor in  C^z and its PORTION T2 where D = 3^7 cause Common Factor in C^z & T2

.......  .............. ....  

[6D] For LEMMA-3 equation C^z = T1 + T2

VERIFICATION of Common Factor in C^z, T2 & T1

..... ......

Lemma-3 based verification of COMMON FACTOR  in C^z & T2 as said in [6C] above

directly implies that for Lemma=3 Terms C^z,  T2 & T1 have COMMON FACTOR 

as given below

T1 = C^z - T2 = K*D - L*D = (K-L)*D where (K-L) in an integer say M where

K = L+M & D > 1 ......... (explained in PART[1B]

which implies that C^z, T2 & T1 have COMMON FACTOR and

Lemma-3 is PROVED 

[Beal Conjecture C^z = A^x + B^y belongs to Lemma-3 equation &

PROOF for Beal Conjecture given from Part [8] onward 

also forms as more detailed PROOF for LEMMA-3 with examples]

... ... .... ....  ...   

[6E] For LEMMA-3 Equation C^z = T1 + T2

Relation Equation of T2 & T1 with C^z

.... ... .....  

As explained in Part[6A] 

RELATION EQUATION of T2 with C^z forms as

T2 = L*D where D is an integer = C^z/K & 

K is an integer such that C^z is divisible by  K

..... .....

Also by the equation T1 = C^z - T2 = K*D - L*D = (K-L)*D

Relation Equations of T1 with C^z form as

T1 = M*D where M is an integer such that M = K- L

.... ..... .. ..... ...... 

[6F] Implications of LEMMA-3 and

(1) Formation of Beal Conjecture Terms A^x & B^y,

(2) REALTION EQUATION Equations of A^x & B^y with C^z  and

(3) PROOF for COMMON FACTOR in BEAL Conjecture terms C^z, A^x & B^y

....... ......

(1) By Lemma-1of Part [1]

for Beal Conjecture C^z = A^x + B^y its terms  C^z & B^y are COMPOSITE numbers such that C & B are integers > 1 where y and z > 2

which implies that 

LEMMA-3 Rectangles are ESSENTIAL CASE Rectangles of Beal Conjecture term C^z ........... ........ 

Also as explained in PART [3] Conditions of Beal Conjecture implies that C^z can be represented by a LEMMA-3 Rectangle for  the equations 

(1) C^z = (C)* C^(z-1),

(2) C^z = C^2*C^(z-2), (3)  C^z = F1*F2 etc 

Also as explained in Part [3] by keeping conditions of Beal Conjecture 

Beal Conjecture term C^z haven't a Rectangle for C^z = 1*C^z

which implies that Beal Conjecture belongs to LEMMA-3 equation

such that 

for proper values of  K, D, L & M in the Lemma-3 based Relation equations

T2 = L/K*(C^z) = L*D form as B^y & T1 =  M*D forms as A^x

Ex: C^z = 9^4

For K = 3, implies D = 3^7  where L =1 & M = (K-L) = 2  give

C^z = K*D = 3*(3^7), T1 = 2*(3^7) & T2 = 1*(3^7)

where for C^z = T1 + T2, the term D = 3^7 cause COMMON Factor in the Terms for C^z, T1 & T2

For the same C^z = 9^4

K = 9 implies D = 9^3, T2 = L*9^3 & T1 = M*9^3 where C^z = 9*9^3 such that

For L = 8 & M = 1 where L+M = K = 9,

T1 = 1*(9^3)  forms as A^x = 3^6 & T2 = 8*(9^3) forms as B^y = 18^3

to give Beal Conjecture 9^4 = 3^6 + 18^3 such that D = 9^3 is COMMON FACTOR in the terms for C^z, A^x & B^y

.... ..... ....  

(2) RELATION Equations of B^y & A^x with C^z

By the same steps that applied to get LEMMA-3 based RELATION EQUATIONS of T2 with C^z

for Beal Conjecture C^z = A^x + B^y where B^y is a portion of Area for Rectangle formed for C^z ,

RELATION EQUATION of B^y with C^z form as

B^y = L*(C^z/K) = L*D where D is an integer = C^z/K such that C^z is divisible by K 

Also as explained for Lemma-3 case for Relation Equation of T1 with C^z

RELATION EQUATION of A^x with C^z forms as

A^x = M*D where M is an integer for K = L+M  and D = C^z/K  such that C^z is divisible by K  

.......... ..... .... 

(3) For C^z = A^x + B^y,

Equation for ONE of the portion of C^z and

Formation of a PORTION of C^z as Beal Conjecture term B^y or A^x

... .... .....

As explained in Part [6B]

For t1 which is a Portion of C^z equation for value of t1 = L/K*C^z where L & K are integers without common Prime Factors 

which implies that

for the values of K such that C^z is divisible by K 

the said portion = L/K*C^z have integer value say T1 such that

T1 = L*(C^z/K) = L*D where T1 and D = (C^z/K) are INTEGERS 

which implies that 

C^z and the said portion T1 can be expressed as Multiples of a same integer = D such that for proper values of K, D and L the term T1 = L*D forms as B^y or A^x

Ex: for C^z = 9^4 value of K = 3 gives C^z/K = D = 3*7 such that

L = 1  gives T1 = L*D = 1*3^7

For the same C^z = 9^4 value of K = 9 gives C^z/K = D = 9^3  such that 

for L = 8 the term T1 = L*D = 8*9^3 forms as B^y = 18^3

.... .... ......  ......   

(4) Based on RECATNGLES formed for  RELATION Equations of A^x & B^y with C^z PROOF for COMMON FACTOR in C^z, A^x & B^y 

..... ........

As Beal Conjecture C^z = A^x + B^y is a LEMMA-3 Equation,

By the same steps applied to prove COMMON FACTOR in C^z & T2 of LEMMA-3 Equation C^z = T1 + T2,

Common Factor in C^z & B^y can be PROVED as given below 

...... ........ ....

In the case of Beal Conjecture C^z = A^x + B^y

For RELATION EQUATION B^y = L*D  where C^z = K*D & D is an integer = C^z/K

such that Composite number C^z is divisible by K and

as explained in Part [6C]

There are 2 CASES for the value of D = C^z/K such that

1.GENERAL CASE (1) where K = C^z and value of  D =1  and

2. ESSENTIAL CASE(2) where K < C^z and value of D > 1

which implies that as explained for LEMMA-3 

For the CASE(2) Directly Equations & Rectangles for B^y = L*D & C^z = K*D verify COMMON FACTOR in C^z & B^y where D >1 cause as COMMON FACTOR and

For the CASE(1) By MODIFIED Rectangles for C^z & B^y  there is verification for COMMON FACTOR in C^z & B^y

such that for C^z/K = D value of K and D varies by keeping inverse proportion 

which implies that

Modified Rectangles of CASE(1) are CASE(2) Rectangles 

Also CASE(1) as said above is GENERAL CASE of PART[6C] where K = C^z &

CASE(2) as said above is ESSENTIAL CASE of PART[6C] where K < C^z

........... ...

(6) Clarification for verification of COMMON FACTOR in C^z and B^y by Relation equation based Rectangles for C^z = K*D and B^y = L*D where D = C^z/K

... ..... ... ......

As explained by PART [3B]

By keeping conditions of Beal Conjecture  

C^z and B^y haven't Rectangles that have breadth =1 for C^z =1*C^z and B^y = 1*B^y such that C^z and B^y can be represented by ESSENTIAL CASE Rectangles that have Length and Breadth to represent integers > 1

which implies that value of D =1 gives the Rectangles formed for Relation Equations where C^z = D*K = 1*C^z  and its related B^y = D*L =1*B^y

where B^y can be represented as an INNER AREA Rectangle for C^z = D*K =1*C^z by sharing the SIDES for D =1 as COMMON SIDE and

there are ESSENTIAL CASE Rectangles for C^z and its Related Rectangles for B^y

where D > 1 such that C^z = K*D and its related B^y =  L*D

that VERIFY Common Factor in C^z and B^y by EQUAL SIDES that represent D > 1

which implies that

as explained in PART[6C] with example for Common factor in 18 and 15

VERIFICATION of Common Factor in C^z & B^y by Essential CASE(2) given above is VALID for ALL Rectangles formed for C^z & B^y including CASE(1) Rectangle

which implies that

Verification of Common Factor in C^z and B^y by ESSENTIAL CASE(2) as said above in this Part PROVES  COMMON FACTOR in C^z & B^y 

therefore

in the case of Beal Conjecture terms C^z & B^y

ESSENTIAL CASE Rectangles formed for RELATION EQUATIONS verify and

PROVE COMMON FACTOR in Beal Conjecture Terms C^z & B^y

where C^z = K*D and B^y = L*D

which also 

PROVE COMMON FACTOR in C^z, B^y & A^x as explained in Part [1B] &

in Lemma3 part

Ex: For C^z = 2^13

K = 2 gives D = 2^12 & L =1 give B^y = L*D = 1*2^12 = 16^3 where D = 2^12 is Common Factor in C^z = 2^13 and B^y = 16^3 which implies

A^x = C^z - B^y =  K*D - L*D = (2-1)*2^12 = 1*2^12 = 4^6

and thus

form Beal Conjecture 2^13 = 4^6 + 16^3 where D = 2^12 cause Common Factor

in related terms for C^z, A^x & B^y 

..... ....... ....... ... .....   

[For 2 Composite numbers T2 = 18 & T1 = 15 where 18 = 15 + 3 

There isn't verification for Common Factor in 15 & 18 by GENERAL CASE Rectangles formed for 18 = 1*18 and Rectangles for 15 =1*15 or 15 = 3*5

There is verification for Common Factor in 15 & 18 by ESSENTIAL CASE Rectangles formed for 18 = 3*6 & 15 = 3*5 by EQUAL SIDES that represents 3 &

that verification by ESSENTIAL CASE Rectangles PROVES T1 = 15 & T2 = 18 have COMMON FACTOR

Similarly as explained in PART[3] by keeping conditions of Beal Conjecture

C^z have ESSENTIAL CASE Rectangles for C^z = C*C^(z-1) and C^2*C^(z-2) that can be modified as one or more Rectangles for C^z = D*K = F1*F2 

where D > 1

which implies that 

Relation Equation based Rectangles formed for C^z & B^y have

Verification of Common Factor in C^z & B^y by ESSENTIAL CASE Rectangles as said above &

PROVES Beal Conjecture terms C^z & B^y have COMMON FACTOR]

......... ....... ... 

Also 

PROOF for COMMON Factor in C^z & B^y directly PROVES COMMON Factor in C^z, B^y & A^x, as explained in the PART [1B]

....... ..... ......  ... ...  ....  

PART [7] SKELETAL Expression based 

VERIFICATION of COMMON FACTOR in C^z & B^y by RECTANGLE with STRUCTURAL figure for C^z  that is explained in Part [5B]

where C^z = U*N = F1*F2 & U = F1 = (t1 + t2) such that

B^y = t1*F2 & A^x = t2*F2

.... ...... ........

As explained in Part [5B]

For STRUCTURAL Figure formed for C^z = U*N = F1*F2

there are F2 number of Unit Rectangles where Unit Rectangle have area = F1

Which implies that

AREA of Rectangle for C^z Can be divided to F2 number of UNIT RECTANGLS where Area of each Unit Rectangle U = F1 that have breadth = 1, Length = F1 &

AREA = 1*F1 = F1

Which represents a SKELETAL EXPRESSION for C^z = U*N = F1*F2 where

C^z = F1 + F1 + F1 + .... ... .... up to F2 number of Unit Terms such that

value of UNIT Term U = F1 & Number of Unit Terms N = F2 where

t1 = B^y/N =  B^y/F2 & B^y = t1*F2 Similarly

t2 = A^x/F2 & A^x = t2*F2

... ... ...   .....   

[7A] For STRUCTURAL Figure for C^z

EXPLANATIONS & steps for VERIFICATION of COMMON FACTOR in C^z & B^y 

are given below & in Part [7B] onward

.... .......

(1) For F1 = t1 + t2 value of t1 can be an INTEGER or a FRACTION such that   

Similar to the CASE OF 7 boxes that have 2 Apples + 3 Oranges in each box that is said in Part[5A]

For Beal conjecture C^z = A^x + B^y where C^z is a Composite number

there are ONE or more STRUTURAL Figure for C^z = U*N = F1*F2 = (t1+t2)*F2

where Unit terms U = F1can be represented as Unit Rectangle of length = F1

such that there are F2 number of Unit Rectangles 

Where A^x & B^y are IDENTICALLY distributed in each Unit Rectangle U = F1 such that F1 = t1 + t2 which implies that

C^z = F1*F2, B^y = t1*F2 and A^x = t2*F2 such that

Number of Unit Rectangles N = F2 cause COMMON FACTOR in Beal Conjecture Terms C^z, A^x & B^y 

as explained in Part [8] onward 

.... ..... ....  .. ...... 

[7B] By keeping Conditions of Beal Conjecture,

for SKELETAL Expression/Structural figure for C^z where C^z = U*N = F1*F2

Minimum value of Number of Unit Terms N  = F2 & Value of Unit term U = F1

.... ......

As explained in  Part [3A] by keeping conditions of Beal Conjecture & for C^z = U*N 

Unit value U = 1 & Number of Unit terms N = 1 are INVALID

such that U & N are integers more than 1 & less than C^z where U = 2^2 or more and

N = 2 or more

Also  as explained in Part [4] 

to verify COMMON Factor in C^z & B^y

for the equation C^z = U*N = F1*F2 where C^z = 1*C^z  = C^z*1

give INSUFFICIENT Rectangles to verify common factor in C^z & B^y

such that

to prove Common factor in C^z & B^y based on SKELETAL Expression/Structural figure or Rectangle for C^z

equation C^z = U*N = 1*C^z where U =1 & N = C^z as well as

equation C^z = U*N = C^z*1 where N = 1 & U = C^z are INVALID 

which implies that

U & N are integers more than 1 and less than C^z such that

Equation C^z = U*N = F1*F2 where U =F1 & N = F2 are 2 factors in C^z give all RELEVANT/valid equations for C^z to PROVE COMMON FACTOR in C^z & B^y

Where U = F1 & N = F2 are INTEGERS more than 1 & less than C^z

such that U = F1 includes U = C^2 & N = F2 include N = C^(z-2)

.... ...... .....   

[7C] For STRUCTURAL Figure of  Beal Conjecture explained in Part [5B] & [7A]

RELATION EQUATION of B^y with C^z 

...... ...... ...

For an integer N > C^z 

Equations C^z/N = U where C^z = U*N = F1*F2 give ALL VALID SKELETAL EXPRESSIONS for C^z = U*N = F1*F2 and its related STRUCTURAL Figures for C^z

where F1 = t1 + t2 and B^y = t1*F2  as explained in Part [7A] 

,,,, ...... 

t1 is PORTION of B^y per Unit Rectangle of area = F1 such that B^y = t1*F2 & 

as a PORTION of F1

t1 = L/K*F1 where L & K are 2 INTEGERS such that L < K  [because t1 < F1 ]

Also for the term L/K meaningless EQUAL FACTORS in Numerator L & Denominator K can be avoided 

which implies that for t1 = L/K*(F1), 

L & K are INTEGRS without COMMON PRIME FACTORS

...... .... ......

Equations t1 = L/K*(F1) & B^y = t1*F2 implies that

B^y = t1*F2 = (L/K*F1)*F2 = L/K*C^z  where  L & K are integers without common prime factors & F1*F2 = C^z

Which implies that

To equate with INTEGER value of B^y where B^y = L/K*C^z

for its RHS part L/K*C^z the term C^z  must be divisible by K 

such that C^z/K = D where D in an INTEGER and

B^y = L*D

... ..... ... 

Thus RELATION EQUATION of B^y with C^z forms as

B^y = L*(C^z /K) = L*D where D is an INTEGER & D = (C^z/K) as explained above

Also C^z/K = D gives C^z = K*D 

which implies that

C^z & B^y can be represented by Rectangles for C^z = K*D & B^y = L*D

such that

B^y can be represented as a INNER AREA Rectangle to Rectangle formed for C^z 

by sharing the SIDE for D such that SIDE for D form as COMMON SIDE (Equal side) 

for the Rectangles for C^z & B^y

.... ...... ...  ..  ....  

PART [8] By STRUCTURAL Figure for C^z where C^z = D*K = F1*F2 = (t1 + t2)*F2 & its

Relation Equation where B^y = L/K*C^z = L*D & D = C^z/K

PROOF for COMMON FACTOR in C^z & B^y 

.... .... ..... ..... .... ,,, ......  .... ...... ........ 

[8A] For Relation Equations C^z/K = D where C^z = D*K,

based on different value of K value of D varies by keeping inverse proportion and

generally there are 2 CASES in VERIFICATION of COMMON FACTOR in C^z & B^y such that

CASE(1) where K = C^z  that give D =1 & C^z = D*K = 1*C^z

CASE(2) where K < C^z  give D > 1

as explained below

...... ....... ... 

CASE(1): For C^z/K = D and its Skeletal Expression 

K = C^z implies Number of Unit terms = C^z and value of Unit term D = 1

As given in [3A] for Skeletal Expression of C^z that keep conditions of Beal Conjecture 

Value of Unit Term and Number of Unit terms must be integer >1

which implies that for CASE(1)

there isn't a SKELETAL Expression/Structural figure  for C^z that keep conditions of Beal Conjecture and Valid to verify Common Factor in C^z and B^y

..... .... .....

Also as explained by Part[4A]

Rectangle formed for C^z = D*K = 1*C^z is INSUFFICIENT to verify COMMON FACTOR in C^z & B^y  &

By RULES in Geometry equation C^z = 1*C^z need to be MODIFIED as C^z = F1*F2

to VERIFY COMMON FACTOR in C^z & B^y

such that for proper values of K < C^z 

Rectangle for C^z = 1*C^z get modified where Modified Rectangles are formed for C^z = D*K = F1*F2

where F1 & F2 are 2 FACTORS in C^z & Every Factor in C^z can be projected to complete VERIFICATION for COMMON FACTOR in C^z & B^y

which implies that

CASE(2) where K < C^z is sufficient to complete VERIFICATION of  COMMON FACTOR in C^z & B^y

such that Case(2) verification of common factor in C^z & B^y are VALID also for CASE(1) 

.... ..... ......  ....   

[8B] VERIFICATION of COMMON FACTOR in C^z & B^y For the CASE(2) as said above

which also include VERIFICATION for CASE(1) by Modified rectangles/Skeletal Expressions for C^z

such that C^z = 1*C^z get modified as C^z = F1*F2

..... ...... ... 

CASE(2): where K < C^z

Based on value of K > C^z  where C^z is DIVISIBLE by K  & D > 1

there are 3 Sub CASES as given below

Sub CASE(2A):  Value of K < F1

Sub CASE(2B): Value of K = F1

Sub CASE(2C): Value of K > F1 but K < C^z

Where D = C^z/K , C^z = D*K & B^y = L*D

...... ..... ......   

For Sub CASE(2A) where K < F1 and (F1*F2) = C^z is divisible by K implies that

B^y = L/K*(F1*F2) = L*(D1)*F2 where (D1) & L are INTEGERS 

such that (D1) = F1/K where C^z = F1*F2 is divisible by K

Also by Part[8A] we have B^y = t1*F2 such that

B^y = L*(D1)*F2  & B^y = t1*F2 which implies that

t1 = L*(D1) and t1 is an INTEGER 

Therefore Sub CASE(2A) have verification for COMMON FACTOR in C^z & B^y  caused by  number of Unit Terms = F2 

where C^z = F1*F2 & B^y = t1*F2 = L*(D1)*F2

.... ... .....

Similarly for Sub CASE(2B) where K = F1

B^y = L/K*(F1*F2) = L*F2 such that

Equations B^y = L*F2 & B^y = t1*F2 implies that t1= L & t1 is an INTEGER

Therefore  Sub CASE(2B) have verification for COMMON FACTOR in C^z & B^y caused by Number of Unit terms F2  

where C^z = F1*F2 & B^y = t1*F2 = L*F2

....... ....... .... .....   

[8C] For Sub CASE(2C) where K > F1 & K < C^z 

Verification of COMMON FACTOR in C^z & B^y is given below

.... ..... ...

For B^y = L/K*C^z where L & K are integers without Common prime Factors implies that 

C^z is divisible by K such that K > F1 and K <  C^z which implies that

C^z/K = P where  P is an integer > 1

Therefore 

for Sub CASE(2C)

B^y = L/K*(C^z) = L*(C^z/K) = L*P where P is an INTEGER > 1

Also C^z/K = P gives C^z = P*K  such that

Equations C^z = K*P* & B^y = L*P  where P> 1 implies that 

for Sub CASE(2C)  C^z & B^y have Rectangles formed for C^z = K*P & B^y = L*P 

that verify Common Factor in C^z & B^y by the EQUAL SIDES that represent the term P > 1

.... ... ..... ....

Also the said value of  K > F1 and K < C^z, say K1 that give 

integer value to the term L*(C^z/K) the equation  B^y = L*(C^z/K) such that

 L*(C^z/K)   = L*D, implies that C^z/K1 = P where P is an integer > 1 and P  < C^z 

where P is a factor in C^z such that C^z = P*(K1)

For C^z = P*(K1) there is a Skeletal Expression & Structural Figure for C^z = U*N = P*K1 such that

C^z = P + P + P + .... .... .... K1 Number of  UNIT terms/Rectangles and there is verification for Common Factor in C^z & B^y as said in the CASE(2A) or (2B)

which implies that

as explained above by CASE(1) & CASE(2) where CASE(2) have Sub CASES (2A), (2B) and (2C)

Structural figure representation for Beal Conjecture term C^z PROVES

COMMON FACTOR in C^z & B^y

.... ..... ....  

[8D] Based on PROOF for COMMON FACTOR in C^z & B^y

PROOF for COMMON FACTOR in C^z, B^y & A^x

... ..... .....

As explained in Part [1B] for Beal Conjecture C^z = A^x + B^y

PROOF for Common Factor in C^z & B^y also PROVES Common Factor in 

C^z, B^y & A^x  

..... ...... .......    

PART [9A] For C^z = F1*F2, where F1 & F2 are 2 factors in C^z

BIFURCATION of ONE of the FACTOR in C^z to 2 INTEGERS and

formation of Beal Conjecture C^z = A^x + B^y

... ....   ....

By Rectangle based verification of Common factor in C^z & B^y as well as

Skeletal Expression based verification of common factor in C^z & B^y

For C^z = F1*F2 Bifurcation of ONE of the Factor in F1 or F2 and then Expansion cause formation of Beal Conjecture C^z = A^x + B^y such that

the Factor which is NOT bifurcated cause Common Factor in C^z, B^y and A^x as given below

Equation C^z = 1*C^z is INSUFFICIENT to verify Common Factor in C^z & B^y

For equation C^z = F1*F2 let F1 is bifurcated to 2 INTEGERS as F1 = L+M

which implies an expansion as

C^z = F1*F2 = (L+M)*F2 = L*F2 + M*F2 = T1 + T2 where T1 & T2 are 2 integers that have  COMMON FACTOR caused by F2 such that

for proper values of L, M & F2 the term T1= L*F2 and T2 = M*F2 form as B^y and A^x 

Ex: For C^z = 9^4 & equation C^z/K = D where C^z = D*K

K = 3^7 give D = 3 where 9^4 = D*K = F1*F2 = [3]*3^7  & for 3 = 1+2 expansion form as

9^4 = (3)*3^7 = (1+2)*3^7 = 3^7 + (2*3^7) where 

C^z = 9^4, T1 = (1*3^7) & T2 = (2*3^7) have COMMON FACTOR caused by F2 = 3^7

.... ....... ....

For the same C^z = 9^4 value of K = 9^3 give D = 8 where for D = L+M  9 = 1+8 give VALID terms for A^x & B^y as given by expansion

9^4 = [9]*9^3 = F1*F2 = [1+8]*9^3 = 1*9^3 = 8*9^3 where T1 =1*9^3 & T2 = 8*9^3 form as

A^x = 3^6 & B^y = 18^3 such that for Beal Conjecture 9^4 = 3^6 + 18^3. where 

C^z = 9^4, A^x = 3^6 & B^y = 18^3 have COMMON FACTOR caused by F2 = 9^3

..... ....... .... . .... ...  

PART [9B] Conditions/Particularities of Beal conjecture terms that cause Common Factor in C^z, B^y & A^z

...... ...

As explained in PART [3]

By keeping conditions of Beal Conjecture for

SKELETAL Expression & STRUCTURAL Figure of Beal Conjecture Term C^z,

Value of Unit term  U & Number of Unit Terms N must be INTEGER more than 1 & less than C^z 

which implies that

Beal Conjecture confirms ESSENTIAL CASE Rectangles and its Structural Figures for C^z such that

for K < C^z Equation C^z/K = D where D > 1 

Which particularity cause COMMON FACTOR in C^z, B^y & A^x such that

B^y = L*D and A^x = M*D where L + M = K

Also by keeping conditions of Beal Conjecture equation C^z = U*N = 1*C^z 

where 

value of Unit term U = 1 as well as equation C^z = U*N = C^z*1 where Number of Unit terms N = 1 are IRRELEVANT equations to Prove RULES related Beal Conjecture

Besides

By Rules geometry Equation C^z = U*N = 1*C^z is INVALID to  verify COMMON FACTOR in C^z & B^y 

Thus as explained in part [9A] for the equation C^z = F1*F2 

there is bifurcation of ONE of the Factor to 2 INTEGERS that cause formation of the equation C^z = A^x + B^y  such that caused by the remaining factors in C^z that are kept unchanged

there is GUARANTEED COMMON FACTOR in C^z, B^y & A^x as explained above in Part [9A]

..... ...... ..... 

[9C] More Examples  

of formation of VALID Terms in Beal Conjecture C^z =  A^x + B^y &

Common Factor in C^z, B^y & A^x

..... ......

Example [2]

For C^z = 2^13 expansion for C^z = F1*F2 = (L+M)*F2 forms as

2^13 = (2)*(2^12) = (1+1)*2^12 + 1*2^12 + 1^2^12 = 4^6 = 16^3 & give

Beal Conjecture 2^13 = 4^6 + 16^3 where F2 = 2^12 cause COMMON FACTOR in the terms for C^z, A^x & B^y

Example [3]

For F1 =  8 bifurcated as 8 = 1+7  & F2 = 7^12 = 49^6

we have C^z = [2*(49^2) ]^3,  A^x = 1*49^6 & B^y = 7*49^6 = 7^13 where Number of Unit terms F2 = 7^12 cause COMMON FACTOR in C^z, A^x & B^y in Beal Conjecture [2*(49^2) ]^3 = (49)^6 + 7^13

Related expansion for C^z = F1*F2 = (L+M)*F2 forms as

[2*(49^2) ]^3 = (2^3)*(49)^6 = (1+7)*(49)^6 = 1*(49)^6 + 7*(49)^6 = 49^6 + 7^13

..... ....... ....

Also 

similar to the case explained in Part [5A] where Apples & Oranges identically placed in 7 Boxes where Number of BOXES cause Common Factor,

For Structural Figure representation of Beal Conjecture term C^z

Number of Unit Term/Rectangles causes COMMON FACTOR in C^z, A^x & B^y as explained below

For the STRUCTURAL Figure formed for C^z = U*N = F1*F2

equation B^y = t1*F2 where t1 = L*(C^z/K) and C^z must be divisible by K

where t1 is an integer implies that 

Basic Unit terms of value 1 in B^y & A^x are IDENTICALLY distributed in F2 number of Unit Rectangles of Area = F1 such that F1 = t1 + t2 

where t1 is an integer as verified by Part [8] such that

for t1 an integer also t2 is an integer where t1 + t2 = F2 and

for proper values of t1, t2 & F2

there are VALID Terms for A^x & B^y where 

C^z = F1*F2, B^y = t1*F2 & A^x = t2*F2 have COMMON FACTOR caused by Number of Unit Rectangles F2 in the System

.... .......  .....  ...

PART [10] Based on GENERAL equation A^x + B^y = K where K is an INTGER,

FORMULA to get  valid terms in Beal Conjecture A^x + B^y = C^z

...... ...... 

There are infinite numbers that can be expressed as A^x + B^y = K where K is an integer

 A^x + B^y = K can be  modified by multiplying all the 3 Terms by suitable integers like K^(x*y) to give VALID terms for Beal Conjecture C^z = A^x + B^y such that

K *K^(x*y) =  K^[(x*y)+1] forms as the term for C^z

A^x get modified as (A*K^y)^x, &

B^y get modified as (B*K)*x)^y

Thus give Beal Conjecture K^[(x*y)+1] = (A*K^y)^x + (B*K)*x)^y

Ex: For 27 + 1 = 3^3 + 1^4 = 28 multiplication by 28^(x*y) = 28^(3*4) = 28^12 gives

Beal Conjecture [3*(28^)^4]^3 + [28^3]^4 = 28^13 

..... ..... ......  

PART [11] PROOF for Beal Conjecture

..... .......

Based on PROOF for COMMON FACTOR in C^z, A^x & B^y 

that is already obtained in this article, 

BEAL CONJECTURE can be PROVED as follows

....... ...... ......

For 6^z the term 6^z = 6*6*6* ..... ..... up to z number of terms which implies that

the term 6^z & its base 6 have same prime factors 

similarly

For Beal Conjecture C^z = A^x + B^y 

C & C^z have the same PRIME FACTORS,

A & A^x have the same PRIME FACTORS, and

B & B^y have the same PRIME FACTORS

which implies that

Prime factors of the COMMON FACTOR in C^z, A^x & B^y cause COMMON PRIME FACTOR in C, A & B 

Therefore Proof for Common Factor in C^z, A^x & B^y 

also proves C, A & B have COMMON PRIME FACTOR and

Proof for COMMON PRIME FACTOR in C, A & B is the needed PROOF for Beal Conjecture

Hence based on PROOF for COMMON FACTOR in C^z & B^y that 

PROVE COMMON FACTOR in C^z, A^x & B^y

this article have PROOF for BEAL CONJECTURE 

..... ..... .....  

PART [12] CONCLUSSION 

...... .....

By main STEPS & VERIFICATIONS  given below, this Article have PROOF for Beal Conjecture

...... ..... 

[1] As explained in PART[1]

For Beal Conjecture,  C^z  is Composite number & Either one among A^x & B^y is COMPOSITE Number such that

along with C^z for this article B^y is taken as COMPOSTE Number 

[2] As explained by PART [3]

By keeping conditions of Beal Conjecture there aren't Rectangles and Skeletal Expressions for C^z = U*N = 1* C^z and B^y = U*N = 1*B^y

By keeping conditions C^z and B^y form as Rectangles that that have both sides to represent integers > 1 for C^z = U*N = C^2*C^(z-2) and B^y = U*N = B^2*B^(y-2)

where C and B are integers >1 which can be modified to other Rectangles that have both sides to represent integers >1 

[3] As explained by PART[6] onward

PROOF for COMMON FACTOR in C^z, B^y & A^x based on

RELATION Equation of B^y with C^z where 

equation C^z/K = D give C^z = K*D and as a portion of C^z

Relation equations give B^y  = L*D and A^x = M*D

such that for the said Rectangles for C^z = D*K and B^y = D*L

besides the General Case Rectangles where D = 1,

as a composite number that keep conditions of Beal Conjecture there are Rectangles for C^z = D*K and 

its modified Rectangles where C^z = D*K = F1*F2 where D >1 such that

B^y is a portion of C^z and

as a portion B^y have Rectangles for B^y = D*L where C^z and B^y are multiples of the same term D > 1

that VERIFY and PROVE Common Factor in C^z, B^y and A^x

[4] PROOF for COMMON FACTOR in C^z, A^x & B^y implies that

C, B & A have COMMON PRIME FACTOR which is the needed PROOF for Beal Conjecture & hence

in this ARTICLE Beal Conjecture is PROVED

..... ........