PROOF for BEAL CONJECTURE from PROOF for COMMON FACTOR in C^z, A^x & B^y (mainly by GEOMETRY)

 INTRODUCTION:

As generalization of FLT in number theory

Beal Conjecture is formulated in 1993 by amateur Mathematician Andrew Beal [USA]

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

if A, B, C, x, y & z are positive integers where x, y & z are all greater than 2

then A, B & C have COMMON PRIME FACTOR 

Ex:  

(1) 9^4 = 3^6 + 18^3 & its modified form 72^4 = 12^6 + 288^3

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

.... .......

Formula to get valid terms for C^z, A^x & B^y in Beal Conjecture 

There are infinite numbers that can be expressed as A^x + B^y = K or

A^x + K = C^z where K is an integer 

Then By multiplying K, A^x & B^y by K^(x*y) or K, A^x & C^z by K^(x*z) valid terms for Beal Conjecture C^z = A^x + B^y are formed 

Ex: A^x + B^y = K  Then multiplying all the 3 terms by K^(x*y) give Beal Conjecture where C^z = K^[(x*y)+1]

A^x = (A*K^y)^x & B^y = (B*K^x)^y

Ex: For A^x + K = C^z  equation 8 = 1+ 7 where 8 =2^3 & 1 = 1^4 gives

C^z = 2^3*(7^4)^3 = [2*(49^2)]^3, A^x = 1*(7^12) = 49^6 & B^y = 7*(7^12) = 7^13

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

Beal conjecture is yet to be proved

.... .....  .....

ABSTRACT:

In this article 

Firstly a PROOF for COMMON FACTOR in Beal Conjecture terms C^z, A^x & B^y is given,  

which will DIRECTLY PROVE Common PRIME FACTOR in C, A & B 

(as explained the following PARTS [IA] onward) 

where  PROOF for COMMON PRIME FACTOR in C, A & B is the PROOF for BEAL CONJECTURE 

.... ... ....

[IA] In this article PROOF for BEAL CONJECTURE is obtained by 4 SIMPLE STEPS as given below:

STEP [1]: Firstly VERIFICATION that give 

C is an integer > 1 and Either One among A & B is integer > 1 

which implies that by taking A as the term that have value > 1 in A & B,

C^z & A^x are COMPOSITE NUMBERS &

STEP[2]: PROOF for COMMON FACTOR in Composite numbers C^z & A^x is obtained by basic methods in GEOMETRY like comparing Length of SIDES of Rectangle or Square figure formed for C^z & A^x

STEP[3]: A PROOF for COMMON FACTOR in C^z & A^x DIRECTLY give PROOF for COMMON FACTOR in C^z, A^x & B^y as given below:

...... ....

If C^z & A^x have COMMON FACTOR  & Let D  is the COMMON FACTOR in C^z & A^x 

it implies that D is an INTEGER > 1 such that  A^x = K1*D & C^z = K2*D

where K1 & K2 are INTEGERS such that K2 > K1   (since C^z > A^x)

Also B^y = C^z - A^x which implies that

B^y = K2*D - K1*D = (K2 - K1)*D  where K2-K1 is an INTEGER (say K3)

which implies that

PROOF for COMMON FACTOR in C^z & A^x also PROVE that C^z, A^x & B^y have COMMON FACTOR

where C^z = K2*D, A^x = K1*D & B^y = K3*D

STEP [4]:  For C^z = A^x + B^y

C^z & C have SAME PRIME FACTORS, similarly

A^x and A have SAME PRIME FACTORS & 

B^y and B have SAME PRIME FACTORS

which implies that

Prime Factors that are in COMMON FACTOR in C^z, A^x & B^y

cause COMMON PRIME FACTOR in C A & B

Thus

A PROOF for Common Factor in C^z & A^x  directly give

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

PROOF for Common Factor in C^z, A^x & B^y directly give

PROOF for Common Prime Factor in C, A & B

where

Proof for Common Prime Factor in C, A & B is the PROOF for BEAL Conjecture

Which implies that as claimed by ABSTRACT of this article & as explained above

we just need a PROOF for COMMON FACTOR in C^z & A^x to PROVE BEAL CONJECTURE 

as explained in PART [IB] onward

...... ..... ... 

[IB] For BEAL Conjecture C^z = A^x + B^y

MINIMUM Values for C, C^z & either ONE of A & B in A^x & B^y

..... ......

(1) Minimum value of C & C^z

For Beal Conjecture C^z = A^x + B^y

A, B, C, x, y, & z are positive integers where x, y & z are > 2

which implies that A^x + B^y > 1^z  

Therefore value of C = 1 is INVALID &  C must be integer > 1 which implies that C^z is a Composite integer 

Also which further implies that 

the least valid term for Beal Conjecture term C^z = 2^3 

..... .... 

(2) Minimum value of  either of  A  & B

For  Beal Conjecture A^x + B^y = C^z we have verified that value of C^z = 2^3 or more 

Therefore for A = 1 & B = 1 value of A^x + B^y = 2 which implies that since minimum value of C^z = 2^3 A & B can't be simultaneously = 1 ( since 1^x + 1^y = 2 )

Thus

We have verified that C & either ONE of A & B are integers > 1 which implies that

C^z & either ONE of A^x & B^y are COMPOSITE NUMBERS and

For this article we take C & A as integers > 1

Thus Based on the verification as given above for this article C^z & A^x are Composite integers &

in this article we have PROOF for COMMON FACTOR in Beal Conjectures terms C^z & A^x where C is an INTEGER  > 1 & A is taken as an INTEGER > 1

which is sufficient to PROVE that C^z, A^x & B^y have COMMON FACTOR as explained in [1A] & in the following parts [IF] onward

..... ......  .... 

[IC] Lemma-1 that prove a structural figure for Beal Conjecture term C^z 

where C^z = [C^2]*[C^(z-2)] 

....... .......  .....

Lemma-1 states that

By keeping conditions of  Beal Conjecture, 

its term C^z have a Skeletal Expression formed by addition of certain number of Unit Term U = C^2 where

C^z = C^2 + C^2 + C^2 + .... .... .... .... up to C^(z-2) Number of Terms such that for the said skeletal expression

value of Unit Term U & Number of Unit terms N are INTEGERS > 1 where U  = C^2 , N = C^(z-2)

which implies that

C^z = U*N = C^2*C^(z-2) 

Also For Beal Conjecture term C^z we have verified that value of C is integer > 1 

which implies that

by keeping conditions of Beal Conjecture its term C^z basically forms as a RECTANGLE that have its both SIDES to represent INTEGERS > 1 for C^z = U*N = C^2*C^(z-2) 

such that C^z = U*N = F1*F2 where U = F1 & N = F2 are 2 FACTORS in C^z

ALSO

without keeping any such conditions & as in the case of all Natural Numbers

value of C^z can be represented as area of a Rectangle for C^z = U*N = 1*C^z

that have breadth to represent U =1 & Length to represent N = C^z itself

which implies a SKELETAL EXPRESSION for C^z as

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

where C^z = U*N = 1*C^z implies U = 1 & N = C^z itself

.... ......  .....  

PROOF  for Lemma-1

(1) In the Series of positive integers every new INTEGER is generated by adding integer 1 to its previous INTEGER in the series which implies that

integer 1 can be taken as BASIC UNIT Term of all positive integers & Every positive integer  have a Skeletal Expression as

T = U + U + U + .... .... ... .... up to N number of  Terms where T = U*N

such that value of Unit term  U & Number of Unit terms N are integers   

Also T = U*N implies a RECTANGLE figure for T that have its BOTH SIDES to represent INTEGERS 

 which further implies that AREA of EVERY RECTANGLE or SQUARE Figure that have its BOTH SIDES to represent INTEGERS values,

implies a SKELETAL EXPRESSION of an INTEGER 

........ ........  

[I D] EQUATION that give  SKELETAL EXPRESSIONS for an integer T &

 2 CASES of value of unit term U such that U = 1 or U >1 & its implied Rectangle or Square Figures for T = U*N which vary for EVERY INTEGER

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

For positive integers T & N such that T is divisible by N

Equation T/N = U implies that 

U is an INTEGER & T is divided to N number of  UNIT Terms of integer value = U which implies

a SKELETAL EXPRESSION for T formed by addition of UNIT TERMS of INTEGER value such that

T = U + U + U + ..... ..... ..... ..... up to N number of Unit Terms where value of Unit term  U & Number of Unit terms N are INTEGERS & T = U*N 

which implies that for T = U*N There are 2 CASES based on value of U

CASE(1) For the value N = T itself Value of U is integer = 1 &

CASE(2) For the value N < T value of U is integer > 1 Where

the CASE(1): T = U*N implies a RECTANGLE figure for T that have breadth to represent integer 1 & Length to represent integer T itself &

The CASE(2) T = U*N implies a RECTANGLE or SQUARE figure that have both SIDES to represent INTEGERS for U > 1  

Also for the CASE(2) where U > 1 & N < T implies that

U & N are 2 FACTORS in T such that T = U*N = F1*F2 where F1 & F2 are 2 Factors in T 

Also the CASE(2) implies T is a COMPOSITE NUMBER 

....... ......  ....  

[IE] By keeping conditions of Beal Conjecture,

SKELETAL EXPRESSION for C^z & its implied RECTANGLE or SQUARE Figures for BEAL CONJECTURE Term C^z

..... ......

By conditions of Beal Conjecture  z is an integer > 2 which implies that

z = 2 + n where n is an integer > 0 

Therefore n = z - 2 &

we have verified that for Beal conjecture C in an integer > 1 which implies that

the least valid term for C^z = 2^(2+1) = 2^3 

Which further implies that by conditions of Beal conjecture,

C^z = C^(2+n) = C^2*C^n = C^2*C^(z-2)

which implies that by keeping conditions of Beal Conjecture

for all values of C & z ,

C^2 can be taken as Unit Term for formation of a Skeletal Expression of Beal Conjecture Term C^z such that

C^z = C^2 + C^2 + C^2 + .... ..... .... up to C^n number of Unit Terms 

where U = C^2, N = C^n = C^(z-2) & C^z = U*N = C^2*C^(z-2)

Also which implies that by keeping conditions of Beal Conjecture

for the SKELETAL EXPRESSION of its term C^z  value of  Unit term U = 1 is RESTRICTED    

which implies that SKELETAL EXPRESSION for C^z that have Unit value U = 1 & Number of Unit terms N = C^z where  C^z = U*N = 1*C^z

and its implied Rectangle figure that have breadth = 1 & Length = C^z itself

are INVALID to represent Beal Conjecture Term C^z especially for the CASES

to PROVE RULES & PARTICULARITIES related to BEAL CONJECTURE term C^z such as Verification of Common Factor in C^z & A^x

Therefore 

By keeping Conditions of Beal Conjecture 

SKELETAL EXPRESSION for C^z forms for C^z = U*N = C^2*C^(z-2) where Unit Term U = C^2 & Number of Unit terms N = C^(z-2)

which implies that as Lemma(1) stated 

basically Beal Conjecture Term C^z forms as a Rectangle Figure that have its BOTH SIDES to represent INTEGERS > 1 for C^z = C^2*C^(z-2) where C in an integer > 1 & z is an integer > 2

...... ........   

[IF] In the case of SKELETAL EXPRESSIONS & its implied RECTANGLE 

For C^z = U*N that keep conditions of Beal Conjecture 

MINIMUM VALUE of  U & N where U = C^2 & N = C^(z-2)

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

By [IB]  For Beal Conjecture term C^z

value of C > 1 & minimum value of valid term for C^z = 2^3 which implies that

for C^z = U*N

Minimum value of U = C^2 is integer 2^2 &

Minimum value of N = C^(z-2) = 2^(3-2) = 2 which implies that

for the least term for C^z onward value of number of Unit Terms N is 2 & more such that

C^z = 2^3 = 2^2 + 2^2 where N = 2,  

C^z =  3^3 = 3^2 + 3^2 +3^2 where N = 3 & so on  

...... ....... .... 

[IG] LEMMA-1 & Conditions of Beal Conjecture implied

SKELETAL EXPRESSION & Rectangle or Square Figure for Beal Conjecture term A^x

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

As explained in [IB]

C^z & A^x are similar terms such that 

C & A are integers > 1 where  x & y are positive integers > 2

which implies that Lemma-1 is applicable also to Beal Conjecture term A^x such that

similar to C^z = U*N = C^2*C^(z-2) where U = C^2 & N = C^(z-2) &

by keeping conditions of Beal Conjecture A^x forms as a Skeletal expression where

A^x = A^2 + A^2 + A^2 + ...... ... ...  up to A^(x-2) terms

that have U = A^2 & N = A^(x-2) such that

A^x = U*N = A^2*A^(x-2) where A & x are positive integers such that A > 1 & x > 2 which implies that Similar to the CASE of  BEAL Conjecture term C^z 

by keeping conditions of Beal Conjecture its term A^x basically have a RECTANGLE or SQUARE Figure that have its both SIDES to represent INTEGER values > 1 for A^x = U*N = A^2*A^(x-2). 

Which further implies that  basically Beal Conjecture terms C^z & A^x form as RECTANGLE or SQUARE Figures that have BOTH SIDES to represent INTEGER values > 1 

....... .....   

[II] VERIFICATION of COMMON FACTOR in Beal Conjecture Terms C^z & A^x along with B^y 

mainly by basic methods & rules in GEOMETRY

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

By Part [I] & LEMMA-1 it is verified that BEAL CONJECTURE Terms C^z & A^x are COMPOSITE numbers &

BASICALLY have figure as 2 RECTANGLES that have its BOTH SIDES to represent INTEGERS > 1 which implies that BASIC METHODS in GEOMETRY can be applied

for VERIFICATION of COMMON FACTOR in 2 composite numbers can be applied to VERIFY COMMON FACTOR in C^z & A^x

.... ...  .... ....

[IIA] METHOD & RULE in GEOMETRY that are applied in this article

to PROVE COMMON FACTOR in Beal Conjecture terms C^z & A^x

where C^z & A^x are COMPOSITE Numbers

..... ...  ....

By RULES in GEOMETRY, 

for Rectangle or Square Figures formed for C^z & A^x  such that  its SIDES represents INTEGER values,

if there are EQUAL SIDES that represent INTEGER > 1, then that EQUAL SIDES verify COMMON FACTOR in C^z & A^x

which implies that  in the case of C^z & A^x where C^z > A^x

Rectangle formed for C^z = 1*C^z that have Breath to represent 1 &  Length to represent C^z itself 

is INSUFFICIENT or IRRELEVANT for VERIFICATION of COMMON FACTOR in C^z & A^x 

since only the factor C^z is projected for C^z for verification & ALL FACTORS in composite integer A^x are LESS than C^z

which further implies that

if C^z is given as Rectangle for C^z = 1*C^z that Rectangle need to be MODIFIED to Rectangle or Square Figures for C^z = F1*F2 where F1 & F2 are 2 FACTORS in C^z 

& every factor in C^z can be projected as F1 or F2 

to VERIFY whether there is COMMON FACTOR in C^z & A^x or NOT 

Example:  For T1 = 18 & T2 = 15

Equations  18 = 1*18 where 15 + 1*15, 15 = 3*5 etc can't verify common factor in 18 & 15 where

18 = 3*6 & 15 = 3*5 are valid terms to verify common factor in 18 & 15 where 3 is exposed as common factor in 18 & 15

which implies that Bigger term  need to expressed as a Multiple of Factor less than that term itself for verification of Common factor with another term less than that term

..... .....  ... ..

[IIB] For verification of common factor in C^z & A^x 

EQUATION that give ALL RECTANGLE or SQUARE Figures of C^z

that have its BOTH SIDES to represent INTEGER values & to complete the verification process 

....... .....  .....

As explained in the Part [ID] Equations T/N = U where T = N*U give Skeletal expression of an integer T & Rectangle or square figure for T 

which implies that

For T = C^z  the same equation  T/N = U forms as C^z/N = U where C^z = N*U &

give Rectangle or Square Figures for C^z  that have its SIDES to represent integer 

which implies that

For proper values of integer N such that C^z is DIVISIBLE by N

the equation C^z/N = U where C^z = U*N gives ALL RECTANGLE or SQUARE Figures for C^z that have its BOTH SIDES to represent INTEGERS &

there are 2 CASES  for C^z = U*N such that 

CASE(1): for the value of N < C^z & value of U >1

which implies that C^z = U*N = F1*F2 where  F1 & F2 are 2 FACTORS in C^z 

CASE(2): for the value of N = C^z value of U = 1 & C^z = U*N = 1*C^z 

Which implies that as explained in the Part [IIA]

the CASE(1): where N > C^z & U > 1 & C^z = U*N = F1*F2 PROJECTS every FACTOR in C^z as the SIDE for U = F1 or N = F2 & verification for COMMON FACTOR in C^z & A^x can be completed &

as explained in [IIA]

CASE(2) where N = C^z, U = 1 & C^z = U*N = 1*C^z 

is IRRELEVANT for VERIFICATION of COMMON FACTOR in C^z & A^x 

which implies that for C^z = U*N 

the CASE(1) where U > 1 is SUFFICIENT to VERIFY COMMON FACTOR in C^z & A^x 

where Rectangles for C^z are formed for C^z = U*N = F1*F2  &

CASE(2) where C^z = 1*C^z  is IRRELEVANT for the said verification process

.......  .......  .....  

[IIC] For the Rectangle formed for Beal Conjecture term C^z = U*N = F1*F2

LEMMA-2 based Relation equation of A^x with C^z &

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

..... ..... ....  

(1) In the case of LEMMA-2,  Beal Conjecture term C^z  is BIFURCATED to 2 INTEGERS such that C^z = T1 + T2 &

LEMMA-2 states that for C^z = T1 + T2

C^z, T1 & T2 are MULTIPLES of a COMMON INTEGER D = C^z/N where N is an integer & C^z is divisible by N

Proof for LEMMA-2 is given below

..... .......  

Lemma-2 based Equation C^z = T1 + T2 implies that

By Geometry  Beal Conjecture term C^z can be represented as AREA of Rectangle formed for C^z  where T1 & T2 can be represented as Bifurcated portions of the Area for C^z &

As ONE of the portion of C^z as said above

T1 = L/N*C^z where L & N are INTEGERS such that L < N [since T1 < C^z]

Also for the fraction L/N  meaningless EQUAL factors in numerator L & denominator N can be avoided which implies that L & N are integers WITHOUT COMMON PRIME FACTORS

which further implies that 

For T1 = L/N*C^z where T1 is an INTEGER

even to equate with integer value of T1 for its RHS part L/N*C^z

C^z must be DIVISIBLE by N

Which implies that 

T1 = L/N*C^z = L*D where D is an integer such that D = C^z/N

Also D = C^z/N implies C^z = D*N

Thus LEMMA- 2 based RELATION EQUATION of  T1 with C^z forms as

T1 = L*D where C^z = N*D such that D is an integer & D = C^z/N where N is an integer & C^z is divisible by N

Therefore C^z = N*D & T1 = L*D implies that 

C^z & T1 are MULTIPLES of an INTEGER =D where D = C^z/N

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

For C^z = T1 + T2 

RELATION EQUATION T1 = L*D where C^z = N*D implies that  T1 &  C^z

can be represented as 2 Rectangles that have BOTH SIDES to represent INTEGER values & 

LEMMA-2 based Rectangles formed for C^z & T1 have EQUAL SIDES that are formed for the term D 

Also 

T1 can be represented as an INNER AREA Rectangle to Rectangle for C^z 

by sharing the SIDE for the term D as COMMON SIDE

Which further implies that the remaining area is a Rectangle for T2 

where T2 = C^z - T1 = N*D - L*D = (N-L)*D &

Rectangle for T2 forms for T2 = (N-L)*D

where N - L is an INTEGER

which implies that

LEMMA-2 based Rectangles for C^z, T1 & T2 have EQUAL SIDES that represent the term D

 ....  .....  ..... 

(2) RELATION EQUATION of T2 with C^z

For C^z = T1 + T2 we have RELATION EQUATION of T1 with C^z as

T1 = L*D where D = C^z/N & C^z = D*N which implies that

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

Thus Lemma-2 based RELATION EQUATION of  T2 with C^z forms as

T2 = M*D where D is an integer = C^z/N  & C^z = N*D

..... ..... ..... ...  

(3) For Lemma-2 equation C^z = T1 + T2 

Implications of Relation Equations T1 = L*D, T2 = M*D where C^z = N*D

..... ...

T1 = L*D & T2 = M*D where C^z = N*D  & D is an integer implies that

As CLAIMED by LEMMA-2

For C^z = T1 + T2 the terms C^z, T1 & T2 are MULTIPLES of a COMMON INTEGER D = C^z/N such that N is an integer & C^z is divisible by N

Ex: for C^z = 9^4 = 6561

For N = 3 & L = 1  T1 = L*C^z/N = L*D = 1*2187 where D =  6561/3 = 2187 &

for N = 3 & M = N -L = 2  implies that T2 = M*D = 2*2187 = 4374 where

T1 = 1*2187, T2 = 2*2187 & C^z = 3*2187 are Multiples of D = 2187

..... ....... ...  ...

(4) LEMMA-2 based Relation Equations of T1 & T2 with C^z & its implied

EXPANSION for  C^z = T1 + T2

T1 + T2 = L*D + M*D = (L+M)*D = N*D = C^z  where M = N-L & N = L + M

implies that expansion for C^z = T1 + T2 forms as

C^z = N*D = (L+M)*D = L*D + M*D = T1 + T2 where N, L & M are integers such that N = L +M

which further implies that

For Equation  C^z/N = D where C^z = N*D the Term C^z is DIVIDED to N number of Terms such that

C^z = D + D + D + .... ..... .... up to N number of terms where

For BIFURCATION of C^z to 2 INTEGERS value N = 1 is INVALID 

which implies that in the case of  BIFURCATION of C^z to 2 INTEGERS

for the equation C^z/N = D & C^z = D*N value of N > 1 & N is a FACTOR in C^z

Therefore

By expressing C^z as MULTIPLE of a FACTOR in C^z  say F1 &

by BIFURCATING that FACTOR to 2 INTEGERS such that F1 = L+M &

then expanding C^z

terms T1 & T2 in LEMMA-2 based equation C^z = T1 + T2 are formed

(5) LEMMA-2 based Relation Equations & Expansion for C^z = T1 + T2

implies that 

T1 can form as a INNER AREA RECTANGLE in the AREA of RECTANGLE formed for C^z by SHARING EQUAL SIDES that represent the term D such that 

the REMAINING AREA is RECTANGLE for T2 such that

T2 = C^z - T1 = N*D - L*D = (N-L)*D  where SIDES that represent D form as EQUAL SIDES for the RECTANGLES for C^z, T1 & T2

which further implies that 

there are 2 CASES for LEMMA-2 Equation C^z = T1 + T2 as given below

CASE(1):  value of N = C^z where value of D = 1 &

CASE(2): value of N < C^z where value of D > 1

For the CASE(1) where D = 1

C^z = D*N = 1*C^z & T1 = D*L = 1*T1 which implies that

C^z forms as a RECTANGLE that have Breadth = 1 & Length = C^z itself &

its AREA is BIFURCATED to 2 portions that have INTEGER value such that

C^z = T1 + T2  where T1 & T2 are INTEGERS 

For the CASE(2) where D > 1 implies that C^z = D*N = F1*F2 where F1 & F2 are 2 FACTORS in C^z &

C^z = D*N = F1*F2, T1 = D*L = F1*T1 & T2 = F1*M where F2 = L+M

which implies that

C^z forms as a RECTANGLE that have both SIDES to represent INTEGERS > 1 for C^z = D*N = F1*F2 &

SIDE for F2 is BIFURCATED to INTEGERS such that F2 = L+M &

in the AREA of RECTANGLE for C^z = F1*F2

T1 forms as a RECTANGLE that have ONE SIDE = F1 & the other SIDE = L

similarly T2 forms as RECTANGLE that have ONE SIDE = F1 & the other SIDE = M which implies that C^z, T1 & T2 have COMMON FACTORS that is VERIFIED by SIDES that represents the term D = F1

Therefore 

CASE(1) where D = 1 & CASE(2) where D > 1 implies that

for the CASE(1)

C^z forms as RECTANGLE for C^z = D*N = 1*C^z &

by BIFURCATING the SIDE for N = C^z to 2 INTEGERS such that C^z = L + M,

the terms T1 & T2 can BIFURCATE AREA of  RECTANGLES for C^z

such that C^z = T1 + T2, T1 = L*D & T2 = M*D 

where SIDE for D = 1 form as COMMON SIDE & BREADTH of  Rectangles for C^z,  T1 & T2

which implies that

L & M can be any 2 INTEGERS among 1 to (C^z - 1) such that L + M = C^z

which implies that

T1 & T2 can be any 2 INTEGERS among 1 to (C^z -1) such that T1 + T2 = C^z &

..... ...... ....  

For the CASE(2)

C^z forms as a RECTANGLE that have BOTH SIDES to represent INTEGERS >1 for C^z = D*N = F1*F2 where D = F1 & N = F2 are 2 FACTORS in C^z & 

by BIFURCATING the SIDE for N = F2 to 2 INTEGERS such that F2 = L+M

the terms T1 & T2 can BIFURCATE AREA of RECTANGLE for C^z to 2 INNER AREA RECTANGLES for T1 & T2 such that

C^z = T1 + T2, T1 = L*D & T2 = M*D where SIDES for D > 1 form as EQUAL side & COMMON SIDE for the RECTANGLES for C^z, T1 & T2 

which implies that 

For the LEMMA- 2  based Equation C^z = T1 + T2 

CASE (2) where N < C^z & D > 1

C^z, T1 & T2 have COMMON FACTOR 

Therefore 

for the CASES where Beal Conjecture term C^z is represented as RECTANGLE for C^z = F1*F2 & its area is BIFURCATED to 2 INTEGERS such that C^z = T1 + T2,

there is COMMON FACTOR in C^z, T1 & T2

which is the CASE of Beal Conjecture C^z + A^x + B^y as explained in PART [IID} onward

..... ......   ....  ...

[IID] BEAL CONJECTURE C^z = A^x + B^y as LEMMA-2 Equation &

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

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

(1) LEMMA-2 Based Expansion of equation C^z = T1 + T2  &

Formation Beal Conjecture  where T1 + T2 form as A^x + B^y

..... ...

Beal Conjecture C^z = A^x + B^y belongs to LEMMA-2 Equation C^z = T1 + T2 

which implies that

based on the expansion C^z = N*D = ( L+M)*D = L*D + M*D = T1 +T2

For proper values of D, L & M the terms T1 = L*D forms as A^x & the term T2 = M*D forms as B^y

Ex:

for N =3, L = 1 & M =2 where N = L+M  value of C^z = 9^4 form as 

9^4 = N*D = 3*3^7 where N= 3 & L+M = 1 + 2  have expansion as

9^4 = 3*3^7 = (1+2)*3^7 = 1*3^7 + 2*3^7 where T1 = 1*3^7 & T2 =2*3^7 which implies that D = 3^7 cause COMMON FACTOR in terms for  C^z, T1 & T2 such that

For proper values of L & D equation  T1 = L*D  give A^x   &

for proper values of M & D give T2 = M*D give B^y 

Ex: For C^z = 9^4 = N*D = (9^3)*(9) where N = 9 is bifurcated for N = L + M such that  9 = 1 + 8 to give VALID Terms for A^x & B^y by expansion as

9^4 = 9*9^3 = (1 + 8)*9^3 = 1*9^3 + 8*9^3

where T1 = 1*9^3 = 3^6 forms as A^x  & T2 = 8*9^3 = 18^3 form as  B^y where D = 9^3 forms as COMMON FACTOR in C^z, A^x & B^y

Thus forms as 9^4 = 3^6 + 18*^3 which can be multiplied by 2^(3*4) to form as another Beal conjecture 72^4 = 12^6 + 288^3

..... ....... ... ... 

[IIE] BEAL Conjecture C^z = A^x + B^y as LEMMA-2 equation C^z = T1 + T2 

 &

RELATION EQUATION of A^x & B^y with C^z

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

By geometry Beal Conjecture C^z = A^x + B^y implies that

C^z can be represented by a RECTANGLE such that A^x can be represented as ONE of the BIFURCATED portion of its AREA & the remaining portion of AREA belongs to B^y

.... ......

(1) RELATION EQUATION of A^x   with C^z

A^x as ONE of the BIFURCATED portion of C^z &

by the same steps applied to get Lemma-2 based RELATION EQUATION of T1 with C^z

As A^x a portion of C^z

A^x = (L/N)*C^z & for L/N where L & N are INTEGERS without COMMON PRIME FACTORS by avoiding meaningless EQUAL factors in numerator L & denominator N  which implies that

to EQUATE with INTEGER value of A^x 

for A^x = L/N*C^z the term C^z must be DIVISIBLE by N &

A^x = L/N*C^z = L*D where D is an INTEGER & D = C^z/N

Thus for Beal Conjecture C^z = A^x + B^y 

LEMMA-2 based RELATION EQUATION of A^x with C^z forms as

A^x = L*(C^z/N) = L*D where L & D are INTEGERS such that C^z is divisible by 

integer N  & D is an integer = C^z/N

Also C^z/N = D implies that C^z = N*D

...... ....  ..... 

(2) LEMMA-2 based RELATION EQUATION of B^y with C^z

B^y = C^z - A^x where A^x = L*C^z/N gives

B^y = C^z - L/N*C^z = C^z*[1 - L/N) = C^z*[N/N - L/N] = C^z*[(N-L)/N] where 

N-L is an integer say M which implies

B^y = M*(C^z/N) = M*D or by direct method

LEMMA-2 based Equation C^z/N = D where C^z = N*D & A^x = L*D implies that

B^y = C^z - A^x = N*D - L*D = (N-L)*D where N - L is an INTEGER say M &

B^y = M*D

Thus LEMMA-2 based RELATION EQUATION of B^y with C^z forms as

B^y = M*D where D is an integer = C^z/N,  C^z = N*D & A^x = L*D

...... ..... 

Ex: for C^z = 9^4 = 6561

A^x = L*(C^z/N) = L*D implies that

For N = 9 & L = 1  one of the bifurcated portion of C^z give VALID term for A^x such that

A^x = L*(C^z/N) = 1*9^3 = 3^6 where D = C^z/N = 9^4/9 = 9^3

which further implies that

for M = N -L = 9 - 1= 8 such that if ONE of the bifurcated portion = 1/9 the other portion = 8/9 where N = L+M 

B^y = 8/9*9^4 =  8*9^3 = 18^3 such that

B^y = M*D = 8*9^3 = 18^3 where D = C^z/N = 9^4/9 = 9^3

Which implies that as claimed by LEMMA-2 

C^z = 9^4, A^x = 3^6 & B^y = 18^3 are MULTIPLES of D = 9^3 &

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

..... .....  ....

(3) Based on LEMMA-2 RELATION EQUATIONS & as BIFURCATION of C^z to 2 INTEGERS,

EXPANSION of C^z = T1 + T2  & formation  Beal Conjecture terms where T1 = A^x & T2 = B^y

..... ....

LEMMA-2 based Relation Equations T1 = L*D, T2 = M*D where D = C^z/N &

 C^z = N*D implies that

T1 + T2 = L*D + M*D = (L+M)*D = N*D =  C^z where N = L+M

Which implies that

as BIFURCATION of C^z to 2 INTEGERS

EXPANSION for C^z = T1 + T2 forms as

C^z = N*D =(L+M)*D = L*D + M*D = T1 + T2 which implies that

Only integer > 1 can be bifurcated to 2 INTEGERS &

by expressing C^z as MULTIPLE of a FACTOR in C^z say F such that C^z = K*F 

where K is an integer &  

BIFURCATING that FACTOR F to 2 INTEGERS such that F = L+M

Equation C^z = K*F = K*(L+M) implies EXPANSION as

C^z = K*F = K*(L+M) = K*L + K*M = T1 + T2  &

for proper values of L, M & K 

T1 = L*K & T2 = L*K form as Beal Conjecture part A^x + B^y 

.... .....

Ex: C^z = 9^4 = K*F = 9^3*9 where K = 9^3 & F = 9  

bifurcation of F = L+M = 1+8 give Beal Conjecture terms A^x = 3^6 & B^y = 18^3 

where the expansion form as

9^4 = 9*9^3 = (1+ 8)*9^3 = 1*9^3 + 8*9^3 = 3^6 + 18^3

Similarly

Beal Conjecture 2^13 = 4^6 + 16^3 have expansion as

C^z = 2^13 = K*F = (2^12)*2 = (2^13)*(1+1) = 1*2^12 + 1*2^12 = 4^6 + 16^3

..... .....  ....  ... 

[IIF] By RECTANGLES formed for LEMMA-2 based RELATION EQUATIONS of A^x & B^y with C^z, 

VERIFICATION of COMMON FACTOR in C^z & A^x along with B^y

where A^x = L*D, B^y = M*D & C^z = N*D such that D is an integer & D = C^z/N

...... .....

As one of the BIFURCATED PORTION of C^z where C^z = A^x + B^y  Relation equations A^x = L*D where D = C^z/N & C^z = N*D

implies that

A^x & C^z have RECTANGLE or SQUARE figures where SIDES that represent the term D form as EQUAL SIDES such that

A^x can form as an INNER AREA RECTANGLE to RECTANGLE for C^z = N*D

by SHARING EQUAL SIDES that represent the term D as COMMON SIDE for RECTANGLES for C^z & A^x

which further implies that

the REMAINING AREA forms as a RECTANGLE that represents B^y 

such that B^y = C^z - A^x  = N*D - L*D = (N-L)*D where (N - L) is an INTEGER

say M & represents LEMMA-2 based RELATION EQUATION of B^y with C^z

where B^y = M*D 

Therefore LEMMA-2 based RELATION EQUATIONS of A^x & B^y with C^z

implies that

as BIFURCATED PORTIONS of Beal Conjecture term C^z

Beal Conjecture terms A^x & B^y can BIFURCATE the AREA of RECTANGLE formed for C^z to 2 INNER AREA RECTANGLES  where C^z = N*D, A^x = L*D & B^y = M*D such that

SIDES that represent the term D form as EQUAL SIDES for Rectangle for C^z & its inner area Rectangle for A^x & B^y

which further implies that for LEMMA-2 based RECTANGLES

formed for C^z = N*D, A^x = L*D & B^y = M*D where SIDES that represent D form as EQUAL SIDES &

there are 2 CASES such that D = 1 or D >1 as explained below

.... ..... ...  

CASE(1): D > 1

D = C^z/N where C^z = D*N implies that for N < C^z value of D > 1  which implies that D & N are 2 FACTORS in C^z such that

C^z = D*N =F1*F2 where F1 & F2 are 2 FACTORS in C^z

Therefore 

for the CASE(1) where D >1

RECTANGLES FORMED for LEMMA-2 based RELATION EQUATIONS

A^x = L*D, B^y = M*D where C^z = N*D

implies that EQUAL SIDES represent integer > 1 & verify COMMON FACTOR in C^z, A^x & B^y 

...... ..... 

CASE(2) D = 1

D = C^z/N where C^z = D*N implies that for N = C^z value of D = 1 which implies that C^z = D*N = 1*C^z &

As explained in PART[IF] by keeping conditions of Beal Conjecture

C^z haven't a RECTANGLE formed for C^z = 1*C^z 

Also as explained by PART[IIA]

equation C^z = 1*C^z as well as RECTANGLE formed for C^z = 1*C^z are IRRELEVANT to VERIFY COMMON FACTOR in C^z & A^x where C^z > A^x such that

if composite number C^z is given as a RECTANGLE for C^z = 1*C^z,

equation C^z =1*C^z & its RECTANGLE need to be MODIFIED to complete verification of common factor in C^z & A^x such that

for N < C^z equation C^z/N = D where C^z = D*N = F1*F2 give the said MODIFIED RECTANGLES of  CASE(2) where C^z = 1*C^z

which implies that 

For Beal Conjecture term C^z

C^z = 1*C^z essentially have MODIFIED RECTANGLES where C^z = D*N = F1*F2

such that

MODIFIED RECTANGLES of CASE(2) RECTANGLE for C^z belongs to CASE(1) RECTANGLES for C^z where D > 1 & there is verification of Common Factor in C^z & A^x as explained in CASE(1) where D > 1

Also verification of COMMON FACTOR in C^z & A^x directly VERIFIES that

C^z, A^x & B^y as explained in PART[IA]

....  ..... ..... 

Ex: C^z = 9^4

N = 9^4 give value of D = 1 where T1 = L*D = L*1 & T2 = M*D = M*1

N = 3  give value of D = 3^7 where T1 = L*3^7  & T2 = M*D = M*3^7 such that D = 3^7 is COMMON FACTOR in C^z, T1 & T2

N = 9 give value of D = 9^3 where T1 = L*D =L*9^3 & T2 = M*D = M*9^3 

which implies that

for the cases N < C^z value of D > 1 &

C^z = N*D, T1 = L*D & T2 = M*D have COMMON FACTOR caused by D > 1 &

for proper values of D, L & M such that D = 9^3, L =1 & M = 8

T1 = L*D = 1*9^3 forms as Beal Conjecture term A^x = 3*6 & T2 = M*D = 8*9^3 forms  as Beal Conjecture term B^y = 18^3

Thus we have Beal Conjecture 9^4 = 3^6 + 18^3

Therefore 

LEMMA-2 based RELATION EQUATIONS of A^x & B^y with C^z are SUFFICIENT to PROVE COMMON FACTOR in C^z, A^x & B^y

where A^x = L*D. B^y = M*D & C^z = N*D such that there are 2 CASES &

CASE(1) where D > 1 directly by RECTANGLES for for C^z = F1*F2 &

CASE(2) where D = 1 by modified rectangles for C^z

VERIFY that Beal Conjecture terms C^z, A^x & B^y have COMMON FACTOR

.... ..... .....  

Explanation for Examples where C^z, A^x & B^y have COMMON FACTOR

For the expansion  C^z = N*D = (L+M)*D = L*D + M*D = T1+ T2 

where T1 + T2 forms as A^x + B^y & D = (C^z/N)

A^x + B^y = L*(C^z/N) + M*(C^z/N) = (L+M)*(C^z/N) = C^z

which implies that (L+M) = N

In the examples given below

the term for N = F2 is bifurcated to 2 integers such that F2 = L + M & 

expansion of C^z  = F1*F2 = F1*(L+M) forms as C^z = F1*L + F1*M = T1 + T2

where C^z is bifurcated as 2 integers that have F1 as COMMON FACTOR &

proper values of L, M & F1 the terms L*F1 + M*F1 form as Beal Conjecture terms

A^x + B^y where C^z, A^x & B^y have  D = F1 as COMMON FACTOR

 ...... ..... 

Ex:  9^4 = 3^6 + 18^3 

for N = 9 where C^z = 9^4  & for C^z/N = D value of D = 9^3 such that

C^z = N*D = F1*F2 = 9^3*9 &

for L + M = N where N = 9 is bifurcated such that L = 1, M = 8 to give

A^x = L*C^z/N = 1*9^3 = 3^6 & B^y = M*C^z/N  = 8*9^3 = 18^3

Thus  C^z = 9^4 = D*N = 9^3*9 have expansion as

9^4 = 9*9^3 = (1 + 8)*9^3 = 3^6 + 18^3  where D = 9^3 forms as common factor

in terms for C^z, A^x & B^y

.... ..... 

Similarly for C^z = 2^13, N = 2 give D = 2^12 such that for N =2, L = 1 & M =1

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

where  D = 2^12 is common factor the terms for C^z, A^x & B^y

Similarly

for C^z = [(2*(49)^2]^3, D = [49^2]^3,  N = 8 such that L = 1 & M = 7

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

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

where D = 7^12 forms as common factor in the terms for C^z, A^x & B^y

Therefore

for C^z represented as a Rectangle for C^z =D*N =  F1*F2  

its BIFURCATED portions A^x = L*D & B^y = M*D are formed by bifurcation of the term for N = F2

which implies that

C^z  & A^x are  MULTIPLES of D = F1 where N > L 

which implies if Beal conjecture term C^z is DIVIDED by A^x  

for C^z/A^x

Numerator C^z is converted as N & Denominator A^x are converted to L where 

N < C^z & L < A^x

which CONFIRMS COMMON FACTOR in C^z & A^x along with B^y

where B^y = M*D

..... ....... ...  

[IIG] SKELETAL EXPRESSION based VERIFICATION for COMMON FACTOR  in Beal Conjecture terms C^z, A^x & B^y

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

In the case of  Beal Conjecture term C^z

For equation C^z/N = D that give C^z = D*N and verification of common factor in C^z & A^x

C^z = D*N implies a Rectangle for C^z as well as implies a SKELETAL EXPRESSION for C^z such that

C^z = D + D + D + ..... ..... up to N number terms where value of Unit term = D & Number of Unit terms = N where N & D are INTEGERS

As explained in PART[IE] 

by keeping conditions of Beal Conjecture

for C^z = D*N value of D = 1 & N = C^z itself are INVALID &

Beal Conjecture implies  C^z = D*N = C^2*C^z-2) such that D = C^2 & N = C^(z-2) where C is an integer > 1 & z is integer > 2 

which implies that

for the least term C^z = 2^3 value of D = 2^2 & value of N = 2 such that

2^3 = 2^2 + 2^2  which implies that 

for Beal Conjecture Term C^z = D*N value of N in an INTEGER 2 or MORE

Ex:  for C^z = 3^3 value of D = C^2 = 3^2 & value of N = C^(z-2) = 3

Also as explained in PART[IIB]

For verification of COMMON FACTOR in C^z & A^x  

For C^z equation C^z = D*N = C^2*C^(z-2) need to be MODIFIED to C^z = D*N = F1*F2  where F1 & F2 are 2 FACTORS in C^z  such that N = F2 is an INTEGER 2 or MORE

which implies that

in the case of verification of  COMMON FACTOR in Beal Conjecture terms C^z & A^x that is based on SKELETAL EXPRESSION of C^z

C^z = D*N = F1*F2 where unit value D = F1 & Number of Unit terms N = F2 implies a SKELETAL EXPRESSION for C^z as

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

where value of Unit Term D = F1 is an integer > 1 & Number of Unit Terms N = F2 is INTEGER 2 or  MORE

.... ...... ....  

[IIH] In the CASE of VERIFICATION of COMMON FACTOR in Beal Conjecture Terms C^z & A^x

EXPANSION for Bifurcation of Beal Conjecture term C^z to 2 integers such that

C^z = T1 + T2  which include Beal Conjecture  C^z = A^x + B^y &

its 2 CASES of Bifurcation of C^z = D*N = F1*F2

.... ..... ..... 

By Lemma-2 based relation Equations

A^x + B^y = L*D + M*D = (L+M)*D = C^z where C^z = N*D 

which implies that N = L+M & expansion for C^z = A^x + B^y forms as

C^z = D*N = D*(L+M) = D*N + D*M = T1 + T2 = A^x + B^y

which implies that 

by expressing C^z as MULTIPLE of a FACTOR in C^z & 

by BIFURCATION of  that FACTOR to 2 INTEGERS

the terms T1 & T2  are formed & for proper values of D, L & M the terms T1 = L*D & T2 = M*D form as A^x + B^y such that

in the case of C^z = 1*C^z  INTEGER integer 1 can't be BIFURCATED to 2 integers

which implies that there are 2 CASES for bifurcation of C^z to 2 INTEGERS & formation of  terms T1 & T2 in the equation C^z = T1 + T2 such that either D = F1 or N = F2 can be bifurcated to 2 INTEGERS as explained below

.... .....

CASE(1): for C^z = D*N & as in LEMMA-2 

number of Unit terms N is BIFURCATED as N = L + M & value of Unit term D is kept FIXED where expansion for C^z = D*N = T1 + T2 forms as

C^z = N*D = (L+M)*D = L*D + M*D = T1 + T2

which implies that T1 & T2 along with C^z are MULTIPLES of Unit term D & for the cases where D>1

there is COMMON FACTOR in C^z, T1 & T2 caused by value Unit term = D &

for proper values of L. M & D the terms T1 & T2 form as A^x & B^y

Ex: C^z = 9^4  & For N = 9  D = C^z/N = 9^3 

where  9^4 = D*N =  9^3*9 implies a SKELETAL EXPRESSION of 9^4 as

9^3 + 9^3 + 9^3 + ..... .... .... up to N number of Unit Terms &

N is bifurcated as N = L + M  & form as 2 SKELETAL EXPRESSIONS such that

9^3 + 9^3 + 9^3 + ...... ...... .... up to L number of Unit terms where L*9^3 forms the integer T1 &

9^3 + 9^3 + 9^3 + ...... .... ..... up to M number of Unit terms where M*9^3 forms as the integer T2

which implies that D = 9^3 cause COMMON FACTOR  in C^z = 9^4, T1 & T2

Ex:

for N = 9, L = 1 & M = 8  C^z = T1 + T2 form as C^z = A^x + B^y where 

T1 + T2 form as 3^6 + 18^3 & Beal Conjecture 9^4 = 3^6 = 18^3 is formed &

D = 9^3 cause COMMON FACTOR in C^z = 9^4, A^x = 3^6 & B^y = 18^3 such that

as explained in PART[IIC(5)] 

value of D = 1 is INVALID for VERIFICATION  of COMMON FACTOR in  Beal Conjecture terms C^z & A^x 

.... ...... ....  

CASE(2): For C^z = D*N 

the term for value of Unit = D is bifurcated such that D = t1 + t2 & 

Number of Unit Terms N is kept FIXED where expansion of C^z = D*N = (t1 + t2)*N forms as

C^z = D*N = (t1 + t2)*N = t1*N + t2*N = T1 + T2 where T1 = t1*N & T2 = t2*N 

which implies that T1 & T2 along with C^z are MULTIPLES of  the term N 

where

by keeping conditions of Beal Conjecture & for verification of Common Factor in C^z & A^x

for C^z = D*N & as explained in PART[IIF] value of  N must be INTEGER 2 or MORE

which implies that

for the cases value of t1 an integer also t2 is integer &

C^z, T1 & T2 have COMMON FACTOR caused by Number of Unit terms  N > 1 & 

for proper values of t1, t2 & N the terms T1 & T2 form as A^x & B^y

...... ..... .... 

Ex: D = C^z/N where C^z = 9^4

for N = 9^3 value of D = 9  which implies a SKELETAL EXPRESSION for 9^4 as

9^4 = 9 + 9 + 9 + ...... .... up to N number of Unit terms where N = 9^3 &

for D = t1 + t2  

C^z = 9^4 gets BIFURCATED as 9^4 = (t1 + t2)^N = t1*N + t2*N = T1 + T2 

where T1 = t1*9^3 & T2 = t2*9^3 implies that

C^z, T1 & T2 are MULTIPLES of N = 9^3 where N must be integer 2 or MORE &

for D = t1 + t2 such that 9 = 1 + 8

T1 = t1*9^3 = 1*9^3 forms as A^x = 3^6  & 

T2 = t2*9^3 = 8*9^3 forms as B^y = 18^3 

Thus give Beal Conjecture 9^4 = 3^6 + 18^3  where Number of Unit Terms N = 9^3 is COMMON FACTOR in C^z = 9^4,  A^x = 3^6 & B^y = 18^3

..... .... .....   

[III] BASIC Concept & GENERAL EXAMPLE 

where 

based on SKELETAL EXPRESSION of an INTEGER T3 or C^z

Number of UNIT TERMS cause COMMON FACTOR in integers T1, T2 & T3 where T3 = T1 + T3

which is applicable to C^z, A^x & B^y in Beal Conjecture C^z = A^x + B^y

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

For a system of 5 IDENTICAL BOXES such that each BOX have 2 APPLES & 3 ORANGES

5, 3 & 2 haven't COMMON FACTOR. 

But for the said system of 5 UNITS,

Number of BOXES (5) forms as COMMON FACTOR in

Number of apples (T1), Number of oranges (T2) & Total Number of Fruits T3 in the system where T3 = T1 + T2 such that

T1 = 2*5, T2 = 3*5 & T3 = (2+3)*5 which implies that

Number of Boxes 5 forms as common factor in T1, T2 & T3 

Similarly

For Beal Conjecture C^z = A^x + B^y where C^z = D*N 

SKELETAL Expression for C^z = D*N must have number of Unit Terms N as an integer 2 or MORE which implies that

for C^z = D*N = F1*F2 where F1 & F2 are 2 FACTORS in C^z

Basic unit terms U =1 in A^x & B^y get identically distributed in F2 number of UNIT Terms in C^z  

where value of each unit term in C^z = F1  is bifurcated as F1 = t1 + t2  & 

For C^z = F1*F2 = (t1+t2)*F2 = t1*F2 + t2*F2 = A^x + B*y 

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

in the case of General example for given above &

as explained in PART [IIIA] onward given below

...... ....... ......  

[IIIA] LEMMA-3 &

Based on BIFURCATION of Unit Term D of SKELETAL EXPRESSION for C^z = D*N such that Unit term D is bifurcated as D = t1 + t2,

VERIFICATION of COMMON FACTOR in C^z, A^x & B^y where  C^z = D*N,

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

where Number of UNIT Terms N cause COMMON FACTOR in C^z, A^x & B^y

..... .... ...... 

LEMMA-3 states that for Beal Conjecture C^z = A^x + B^y

C^z = F1*F2 where F1 & F2 are 2 FACTORS in C^z give all valid Rectangles for C^z to verify common factor in C^z & A^x where

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

that have F1 as area of Unit Rectangle & F2  as number of Unit  Rectangles

where A^x & B^y are IDENTICALLY distributed in each Unit Rectangle such that Number Unit Rectangles N >1 cause COMMON FACTOR in C^z, A^x & B^y

...... ..... ... 

(1)Explanation for LEMMA-3

Equation C^z/N = D & C^z = D*N implies a SKELETAL EXPRESSION for Beal Conjecture term C^z 

where value of Unit Term = D & Number of Unit Terms = N such that

as explained in PART[IIA] value of number of Unit terms N must be integer 2 or MORE

which implies that

C^z = D*N = F1*F2 where F1 & F2 are 2 FACTORS in C^z &

for LEMMA-3

C^z = D*N = F1*F2 implies a SKELETAL EXPRESSION of C^z as

C^z = F1 + F1 + F1 + ..... ..... ..... up to F2 number of terms where D = F1 & N = F2 are INTEGERS > 1

which further implies that

by GEOMETRY AREA of RECTANGLE formed for C^z = F1*F2 can be divided to F2 number of UNIT RECTANGLES where

each Unit Term F1 can be represented as Unit RECTANGLE that have breadth = 1, Length = F1 & Area = 1*F1 = F1

where Number of Unit Rectangles F2 is INTEGER 2 or MORE

Also by geometry  all Unit Rectangles are IDENTICAL

which implies that for C^z =D*N = F1*F2 where C^z = A^x + B^y

Area of each Unit Rectangle of C^z can be IDENTICALLY bifurcated such that

F1 = t1 + t2 where  t1 is portion of A^x per Unit Rectangle of C^z

Therefore t1 = A^x/F2 &  A^x = t1*F2

Similarly the remaining portion in each Unit Rectangle belongs to B^y such that

t2 = B^y/F2 & B^y = t2*F2

which implies that 

for LEMMA-3

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

Number of Unit Rectangles F2 forms as Common term as Multiple in A^x = t1*F2,

B^y = t2*F2 & C^z = F1*F2

which further implies that t1 can be an INTEGER or FRACTION such that

For the CASES where t1 is an INTEGER also t2 is an INTEGER where t2 = F1- t1 &

Number of Unit Rectangles F2 in C^z form as COMMON FACTOR in Beal Conjecture terms C^z, A^x & B^y 

Ex:

For C^z = 9^4 = D*N = F1*F2 =  9*9^3 where D = F1 & N = F2 are 2 FACTORS in C^z such that

9^4 = 9 + 9 + 9 + ..... ..... ..... up to 9^3 number or terms where D = 9 can be represented as UNIT Rectangle of area = 1*9 = 9 where its Breadth = 1 &

for D = 9 bifurcated as t1 + t2 = 1 + 8 

C^z = 9^4 is BIFURCATED as T1 + T2 where T1 = t1*F2 = 1*9^3 form as A^x = 3^6 Also T2 = t2*F2 = 8*9^3 forms as B^y = 18^3

Where  N = F2 = 9^3 is number of Unit Rectangles

Thus to give Beal Conjecture 9^4 = 3^6 +18^3 where  N = 9^3 cause COMMON FACTOR in C^z = 9^4, A^x = 3^6 &  B^y = 18^3

PROOF for LEMMA-3 is explained in PART[IIIB] onward

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

[IIIB] LEMMA- 3 based  RELATION Equations of A^x & B^y with C^z & VERIFICATION for COMMON FACTOR in C^z, A^x & B^y 

where C^z = F1*F2, F1 = t1 + t2, A^x = t1*F2 & B^y = t2*F2

.... .... ... ... 

[1] Lemma-3 based RELATION EQUATION of A^x with C^z 

where A^x = t1*F2 & C^z = F1*F2

... .. ......

As explained in PART[IIIA] 

For Lemma-3 SKELETAL EXPRESSION for C^z = D*N = F1*F2 

Area of Unit Rectangle = F1 & Number of Unit Rectangles = F2  where F1 is bifurcated as F1 = t1 + t2 such that A^x = t1*F2 &  B^y = t2*F2

where Number of Unit Rectangles N is integer 2 or more

which implies that as a portion of F1

t1 = L/K*F1 where L & K are 2 integers such that L < K ....... [since t1 < F1]

Also 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

Which implies that

A^x = t1*F2 = L/K*F1*F2 = L/K*C^z where L & K are integers without COMMON PRIME FACTOR & C^z = F1*F2

which further implies that

to equate with the INTEGER value of LHS part A^x

for the RHS part L/K*(F1*F2)  the term F1*F2 = C^z must be DIVISIBLE by K 

Thus LEMMA-3 based RELATION EQUATION of A^x with C^z forms as

A^x = L/K*(F1*F2) = L*(C^z/K )= L*D where  D is an INTEGER = (F1*F2)/K such that K is an INTEGER  & (F1*F2) = C^z is DIVISIBLE by K 

.... .... ....  

Also Lemma-2 equation C^z/N = D  where C^z = D*N &

Lemma-3 equation C^z/K = D where C^z = D*K are IDENTICAL such that

As explained in PART[IIG] 

for LEMMA-2 based SKELETAL EXPRESSION  formed for C^z/N = D &  for its equation C^z = D*N value of N must be an integer 2 or MORE which implies that

For LEMMA-3 based equation C^z/K = D & C^z = D*K value of K is an integer 2 or more 

Which implies that 

for K = 1 & K = C^z there isn't VALID Rectangles or Skeletal Expression for C^z to verify Common Factor in C^z & A^x such that

for K = 1 & K = C^z equation C^z = D*K = 1*C^z

Therefor to verify common factor in C^z & A^x  for C^z = D*K value of  K >1 & K < C^z  implies that 

for LEMMA-3 based equation C^z = D*K = F1*F2 where F1 & F2 are 2 FACTORS in C^z  & 

Unit Value D = F1 as well as Number of Unit Rectangles  K = F2  are integers > 1   

which further implies that 

For verification of COMMON FACTOR in C^z & A^x 

based on LEMMA-3 SKELETAL EXPRESSION for C^z = D*K = F1*F2 where A^x = L*D where  D = C^z/K = F1 

There are 2 CASES for value of K > 1 & K < C^z

CASE(1): K < F1 or K = F1 and F1 is DIVISIBLE by K

CASE(2): K > F1 & K < C^z where  C^z is DIVISIBLE by K and C^z = F1*F2 

...... ..... .....  

For the CASE(1) as said above where F1 is divisible by K & t1 = L/K*F1

A^x = L*(F1/K)*F2 implies that A^x = L*P*F2 where P is an INTEGER = F1/K

Also as explained in PART [IIIA] for LEMMA-3 

A^x = t1*F2  & B^y = t2*F2 where t1 + t2 = F1 & C^z = F1*F2 therefore

A^x = t1*F2 & A^x = L*P*F2  where L*P is an INTEGER 

Which implies that t1 = L*P & for A^x = t1*F2 value of t1 is an INTEGER

which further implies that

C^z & A^z have COMMON FACTOR caused by number of Unit Rectangles N = F2

where A^x = t1*F2 = L*P*F2 & C^z = F1*F2

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

Also for F1 = t1 + t2 value of t1 an integer implies that also t2 is an INTEGER &

A^x = t1*F2, B^y = t2*F2 & C^z = F1*F2 implies that A^x, B^y & C^z have COMMON FACTOR where A^x = t1*F2, B^y = t2*F2 & C^z = F1*F2

.... ...

Ex: C^z 9^4 = F1*F2 = 9*9^3 where area of Unit Rectangle F1 = 9 & Number of Unit Rectangles = 9^3

for K = 9  t1 = L/K*F1 = 1 & A^x = 1*9^3 = 3^6

For t1 = 1 value of  t2 = F1-1 = 8 & B^y = 8*9^3 = 2*3*9^3 = 18^3

Thus Beal Conjecture 9^4 = 3^6 + 18^3 is formed where F2 = 9^3 is COMMON FACTOR in C^z = 9^4, A^x = 3^6 & B^y = 18^3

..... ..... ... 

For the CASE(2) as said above where K < C^z & C^z is divisible by K

A^x = L/K*F1*F2 = L/K*C^z = L*S where S = C^z/K & S is an integer > 1 which implies that S is a Factor in C^z 

Also C^z/K = S implies C^z = S*K 

which implies that

A^x & C^z have COMMON FACTOR along with B^y (As explained in PART[IA]

Where B^y = C^z - A^x = K*S - L*S = (K - L)*S & 

K - L is an INTEGER say (K-L) = M & S in an integer > 1 which implies that

C^z, A^x & B^y have COMMON FACTOR 

..... .... ... 

Lemma-3 equations C^z/K = D & C^z = D*K give Skeletal Expression & Rectangle for C^z implies that for different values of K & by keeping value of C^z fixed

Case(1) & Case(2) are MODIFIED Equations & figures of ONE ANOTHER where terms for K & D varies by keeping inverse proportion 

Ex: For C^z = 9^4 = 6561

In the CASE(1) value of K = 9^3 give D = 9 & C^z = (9)*(9^3)

where D = 9 is bifurcated as  9 = 1 + 8  & give

9^4 = (1+8)*9^3 = 1*9^3 + 8*9^3 = A^x + B^y  where A^x = 1*9^3 = 3^6 = 729 &

B^y = 8*9^3 = 5832 such that Number of Unit Rectangles 9^3 forms as COMMON FACTOR in A^x, B^y & C^z

Which further implies that by keeping C^z = 9^4 as fixed

In the CASE(2) value of K = (9^3)/3 = 243 = CASE(1) value of  K/3 where CASE(2) value of D forms as CASE(1) value of D*3 = 9*3 = 27

which implies that 

For CASE(2) there is a LEMMA-3 based Skeletal expression where area of Unit Rectangle = 27 & Number of Unit Rectangle = 243

where CASE(2) Rectangle for 27*243 is a MODIFIED Rectangle of CASE(1) Rectangle formed for 9* 9^3 = 9*729 where

CASE(1) D = 9 bifurcated as 9 = 1+8 which implies that

value of CASE(2)D = value of CASE(1)D*3 &

CASE(2) D = 27 get bifurcated such that CASE(2)D = CASE(1)D*3 = (1+8)*3 = 3+24

which implies that 

CASE(1) expansion for C^z = A^x + B^y forms as

9^4 = 9*9^3 = (1+8)*9^3 = 1^9^3 + 8*9^3 = 3^6 + 18^3 = A^x + B^y & Number of Unit Rectangles 9^3 forms as COMMON FACTOR in C^z, A^x & B^y where

CASE(2) expansion for C^z = A^x + B^y  forms as

9^4 = 27*243 = (3+24)*243 =  3*243 + 24*243 = 3^6 + 18^3 = A^x + B^y & Number of Unit Rectangles 243 forms as COMMON FACTOR in C^z, A^x & B^y

...... ....  ..

Therefore 

For K = 1 & K = C^z equations C^z/K = D give C^z = D*K = 1*C^z which are IRRELEVANT equations to verify Common Factor in C^z & A^x

which implies that 

for K > 1 & K < C^z

verification of COMMON FACTOR in C^z & A^x can be completed by representing C^z = D*K =F1*F2 as Lemma-3 based Skeletal Expression & 

based on equation C^z = F1*F2, Relation equation A^x = t1*F2 = L/K*(F1*F2) & particularity that number of Unit Rectangles must be 2 or more 

there is verification for COMMON FACTOR in C^z & A^y

as explained above in this PART & there are 2 CASES such that by both of the said cases there is verification of COMMON FACTOR in C^ & A^x along with B^y where

value of Number of Unit Rectangles K = F2 which must be integer 2 or MORE cause COMMON FACTOR in Beal Conjecture terms C^z & A^x along with B^y 

.. .... .... 

Also relation equations  A^x = L*(C^z/K) = L*D = t1*F2 where C^z = D*K = F1*F2

implies that for the value of K that give integer value to the term C^z/K = D & valid solution for A^x with integer value

essentially there is one or more LEMMA-3 based Skeletal expression for Beal Conjecture term C^z & Verification for Common factor in C^z & A^x along with B^y as explained in PART[IIIB] given above

....... ......  ...

Ex:  C^z = [(2)*(49)^2]^3 = (8)*(7^12)

For SKELETAL EXPRESSION for C^z 

there are 2 CASES

CASE(1) For D = (7^12) & K = 8 where Bifurcation of of D as D = (7^12) = (L+M) give expansion for C^z = T1 + T2  as

C^z = [(7^12)*8] = (L+M)*8 = (L+M)*8  = L*8 + M*8 = T1 + T2 &

C^z, T1 & T2 have COMMON FACTOR K = 8 where K = 8 is Number of Unit Rectangles

CASE(2) For D = F1 = 8 & K = F2 = 7^12 

Bifurcation of D = 8 as D = L+M where L =1 & M = 7 give t1 = 1*8/8 = 1 & t2 = F1-t1 = 7 where  A^x = 1*F2 = 7^12 = 49^6 & B^y = 7*7^12 = 7^13

Also expansion for C^z = T1 + T2 forms as

[(2)*(49)^2]^3 =  D*K = (8)*(7^12) = (1+7)*7^12 = 1*7^12 + 7*7^12 = (49)^6 + 7^13 &

give Beal Conjecture [(2)*(49)^2]^3 =  49^6 + 7^13 where number of Unit Rectangles K = 7^12 form as COMMON FACTOR in the terms for  C^z, A^x & B^y

..... .....  

Other Examples

[1] Beal Conjecture 9^4 = 18^3 + 3^6

C^z = 9^4 = U*N = F1*F2 = 9*9^3 where F1 = 9 & F2 = 9^3

For K = 9 & L = 8 value of t1 = 8/9*9 = 8 where t2 = 9 - 8 = 1

Thus A^x = t1* F2 = 8*9^3 = 18^3 &  B^y = t2*F2 = 1*9^3 = 3^6 &

F2 = 9^3 is COMMON FACTOR in 9^4, 18^3 & 3^6

Ex:[2]  2^13 = 4^6 + 16^3

For C^z = 2^13 = F1*F2 = 2*2^12 where F1 = 2 & F2 = 2^12

values of K = 2 & L = 1 give t1 = L/K*2 = 1 & A^x = t1*F2 = 1*2^12 = 4^6 such that

t2 = (F1 - t1) = 2 - 1 = 1 & B^y = 1*2^12 = 16^3 &

F2 = 2^12 forms as COMMON FACTOR

....... ........ ......  

[IIIC] Conclusion of PROOF for COMMON FACTOR in Beal Conjecture terms C^z, A^x & B^y where C^z = A^x + B^y

(1) By Conditions of Beal conjecture Basically C^z forms as Skeletal Expression & Rectangle for C^z = F1*F2 where F1 = C^2 & F2 = C^(z-2)

(2) Also by rules to verify common factor in C^z & A^x equation C^z = C^2*C^(z-2) need to be modified as C^z = F1*F2 where F1 & F2 are 2 FACTORS in C^z where C^z = 1*C^z is an irrelevant equation 

(3) Therefore only equation C^z = F1*F2 is relevant for verification of common factor in C^z & A^x

For K < C^z equation C^z/K = D give C^z = D*K = F1*F2 where F1 & F2 are INTEGERS > 1 & 2 FACTORS in C^z

(4) Relation equation A^x with C^z forms as A^x = L*(C^z/K) = L*D  where  D is an integer where C^z/K = D & C^z = D*K which implies that

for K < C^z value of D is a Factor in C^z where D = C^z/K & C^z = D*K = F1*F2 

where A^x = L*F1

which implies that A^x can be expressed as a Multiple of a factor in C^z &

C^z & A^x have common factor

(5) In the case of  Lemma-3 equation A^x = L*(C^z/K) = L*D where K < C^z

For the value of K < C^z that give INTEGER value to the term [L*(C^z/K)] & valid solution for A^x = L*(C^z/K) = L*D 

there is LEMMA-3 Skeletal expression for C^z  &

PROOF for COMMON FACTOR in C^z, A^x & B^y as explained in Part[IIIA] & Part[IIIB]

..... ...... ..... 

[IIID] PROOF for BEAL CONJECTURE

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

For Beal Conjecture term C^z = F1*F2 where F1 & F2 are 2 factors in C^z,

By Geometry the term C^z can be represented as Area of F2 number of Unit Rectangles where Number of Unit Rectangles must be INTEGER 2 or more & Area of Unit Rectangle = F1

Thus

By PART [IIIB] PROOF for COMMON FACTOR in Beal Conjecture terms A^x, B^y & C^z is completed

which implies that as explained in PART[IA] COMMON FACTOR  in A^x, B^y & C^z cause COMMON PRIME FACTOR in A, B & C 

which is the PROOF for BEAL CONJECTURE

Ex: (1) For  9^4 = 18^3 + 3^6 

COMMON FACTOR 9^3 in C^z =  9^4,  A^x = 3^6 & B^z =  18^3 cause COMMON PRIME FACTOR 3 in C = 9,  A = 3 & B = 18 where 3 is COMMON PRIME FACTOR in COMMON FACTOR 9^3

similarly

(2) For 2^13 = 4^6 = 16^3 

common factor 2^12 in C^z = 2^13, A^x = 4^6 & B^y = 16^3 cause

COMMON PRIME FACTOR 2 in C = 12, A = 4 & B = 16

(3) For [2*49^2]^3 = 49^6 + 7^13

common factor 49^6 in C^z = [2*49^2]^3, A^x = 49^6 & B^y = 7^13 cause

COMMON PRIME FACTOR 7 in C = 2*49^2, A = 49 & B = 7

Therefore as CLAIMED & EXPLAINED in PART[IA]

by proving COMMON COMMON FACTOR in C^z & A^x, 

in this article

BEAL CONJECTURE is PROVED mainly by basic methods & rules in GEOMETRY