[IA] ABSTRACT: In 1993 Andrew Beal (USA) formulated & proposed Beal Conjecture which is a generalization of Fermats Last Theorem, where Beal Conjecture states that for C^z = A^x + B^y there is Common Prime Factor in C, A & B
In this article we prove that Beal Conjecture terms C^z, A^x & B^y have COMMON FACTOR which cause COMMON PRIME FACTOR in its bases C, A & B
Therefore in this ARTICLE firstly we PROVE that Beal Conjecture Terms C^z, A^x & B^y have COMMON FACTOR which directly implies that C, A & B have COMMON PRIME FACTOR and thus PROVE the BEAL CONJECTURE
[IB] INTRODUCTION: Beal Conjecture states that if C^z = A^x + B^y where
A, B, C, x, y & z are positive integers such that x, y, & z > 2
then A, B & C have Common Prime Factor
Examples for Beal Conjecture
(1) 2^13 = 4^6 + 16^3 where 2 is Common prime Factor in A = 4, B = 16 & C = 2
Similarly
(2) 72^4 = 12^6 + 288^3 where 3 & 2 are Common prime Factors
(3) [2*(49)^2]^3 = (49)^6 + 7^13 where 7 is Common prime Factor
[IC} Proof for Beal Conjecture
Beal Conjecture claims that for Beal Conjecture C^z = A^x + B^y the base numbers A, B, & C have Common prime Factor & that claim need a PROOF which is known as PROOF for Beal Conjecture
[II] As explained in the following parts
In this article we prove that for Beal Conjecture C^z = A^x + B^y there is Common Factor in C^z, A^x & B^y which cause Common Prime Factor in C, A & B
[IIA] Essential Common Unit Term needed to express 2 entities E1 & E2 as
E1 + E2 = E3 where E1, E2 & E3 are expressed as single terms
Ex(1): For E1 = 1Kg & E2 = 1 pound units Kg & Pound can be converted to same unit & then E3 have a single Term solution as given below
E1 + E2 = 1Kg + 1 Pound = 1 Kg + 0.454 Kg = ( 1 + 0.454) Kg = 1.0454 Kg
Ex(2): For E1 = 1Kg & E2 = 1 meter E3 haven't a single Term solution as Unit Kg & Unit meter can't be converted to same unit
Ex(3): For E1 = 3 & E2 = 5 E3 have a single Term solution as 8 where integer 1 forms as Common Unit Term for 3, 5 & 8 as in 3*1 + 5*1 = (3+5)*1 = 8
[IIB] Integer =1 as Basic Unit Term of integers & Skeletal Expression of integers as Summation of certain number of Unit Terms of integer value = U
In Number series every integer is formed by adding 1 to its previous integer > 0 &
integer 1 can be taken as Basic unit of integers which implies that every integer T have Skeletal Expression formed as
T = U + U + U + ..... .... .... up to N number of Terms where U = value of Unit term = 1 & N = number of Unit Terms = T itself & T = U*N = 1*T
For Composite integers 2 or more Basic unit Term U =1 can be united to give modified Skeletal Expressions where U > 1 as in the case of 6 where 6 = 1+1+1+1+1+1 &
6 = 2+2+2 where U = 2 & N = 3
[IIC] Equation for Skeletal Expressions for an integer T
T/N = U where N is an integer & T is divisible by N implies that T is divided to N number of equal Terms of integer value U & implies a Skeletal Expression for T as
T = U + U + U + .... .... .... up to N number of Terms where U = value of Unit Term
N = number of Unit terms & T = U*N
Based on value of Unit Term there are 2 CASES for Skeletal Expressions of an integer T
Case(1): For N = T itself U = 1 & T = U*N = 1*T
Case(2): For N < T value of U > 1 & for T = U*N = F1*F2
where F1 & F2 are 2 Factors in T
[IID] U-N identity Rectangle or square Figures for integers
T = U*N gives a Rectangle or Square Figure for an integer which implies a Skeletal Expression of integer T
[IIE] Minimum value of important Numbers & Terms related to Beal Conjecture
For Beal Conjecture
A, B, C, x, y, & z are positive integers where x, y & z are > 2
which implies that A^x + B^y > 1^z & for A^x + B^y = C^z
C > 1 & C^z is a Composite integer which implies that
the least valid term for C^z = 2^3
For A^x + B^y = C^z value of C^z = 2^3 or more implies that A & B can't be simultaneously = 1
[IIF] Therefore we have verified that for Beal Conjecture C^z = A^x + B^y
C > 1 & C^z is a composite integer & either A or B must be integer > 1
which implies that A^x or B^y is a Composite integer.
Let we take A > 1 in A & B which makes A^x as the verified Composite integer in A^x & B^y
Thus for this article we have to consider that for Beal Conjecture
C^z & A^x are Composite integers
which is sufficient to PROVE that there is Common Factor in C^z, A^x & B^y
as explained in the following parts where PROOF for Beal Conjecture is obtained directly from the PROOF for Common Factor in C^z, A^x & B^y
[IIG]Particularities of Skeletal Expressions of Beal Conjecture Term C^z & value of its Unit Term in C^z = U*N
For Beal Conjecture z is integer > 2 which implies that
z = 2 + n where n is an integer > 0 & n = z - 2
Also we have verified that C > 1 which implies that
the least valid term for C^z = 2^(2+1) = 2^3 &
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 U for Skeletal Expression of Beal Conjecture Term C^z & forms as C^z = C^2 + C^2 + C^2 + .... ..... .... up to C^(z-2) number of Unit Terms
where U = C^2, N = C^(z-2) & C^z = U*N = C^2*C^(z-2)
Also for Beal Conjecture C > 1 implies that
By keeping conditions of Beal Conjecture Basic Unit U = 1 is restricted &
Skeletal Expression of Beal Conjecture term C^z belongs to CASE(2) Skeletal Expressions where U & N are integers > 1 & < C^z
[IIH] Minimum values for U & N for Beal Conjecture term C^z = U*N &
confirmation for C^z = U*N = F1*F2 where F1 & F2 are 2 factors in C^z
For least valid term C^z = 2^3 Skeletal Expression by keeping conditions of Beal Conjecture forms for C^z = U*N = C^2*C^(z-2) as
2^3 = 2^2 + 2^2 where U = 2^2 & N = 2^(3-2) = 2 = C
For the next higher value of C = 3 Skeletal Expression for 3^3 forms as
3^3 = 3^2 + 3^2 + 3^2 where U = 3^2 & N = 3^(3-2) = 3 = C
For 2^4 = U*N = 2^2*2^(4-2) skeletal expression forms as
2^4 = 2^2*2^2 = 2^2 + 2^2 + 2^2 + 2^2 where U = 2^2 & N = 2^(4-2) = 4
Similarly for 3^4 skeletal expression forms for 3^4 = U*N = 3^2*3^(4-2) where N = 9
which implies that as value of z increases value of N increases such that MINIMUM value of N = 2
Therefore for Beal Conjecture term C^z = U*N value of U & N are integers > 1
which implies that C^z = U*N = F1*F2 where F1 & F2 are 2 Factors in C^z
[III] Lemma-1 & its Rules for Bifurcation of an integer T3 to 2 integers to form the equation T3 = T1 + T2 &
Common Unit Term U of integer value as Multiple in T3, T1 & T2
where T1 = L*U, T2 = M*U & T3 = N*U
Proof:
In Geometry T3 can be represented as area of a Rectangle formed for T3 = 1*T3 &
T3 = T1 + T2 implies that area of the Rectangle for T3 can be bifurcated where One of the portion represents T1 & forms as a Rectangle for T1 = 1*T1 & the remaining area forms as a Rectangle for T2 = 1*T2 where side that represent 1 form as Common side
[IIIA] BASIC RELATION Equation of T1 with T3 in T3 = T1 + T2
As a portion of T3 ( also as a portion of the area that represents T3)
T1 = L/K*T3 where L & K are integers such that L < K (since T1 < T3)
Also L & K are integers without Common Prime Factors by avoiding or by cancelling
EQUAL Factors in Numerator L & Denominator K of the term L/K
which implies that even to equate with integer value of T1
for T1 = L/K*T3 the integer T3 must be divisible by K &
T1 = L*D where D is an integer = T3/K
By taking N instead of K & U instead of D we have Basic Relation Equation of T1 with T3 as T1 = L*(T3/N) = L*U & already explained Skeletal Expression for T3 can be linked with Basic Relation Equation
[IIIB] BASIC RELATION Equation of T2 with T3 where T3 = T1 + T2
We have T1 = L*(T3/N) = L*U where U is an integer = T3/N & T3 = U*N
which implies that
T2 = T3-T1 = T3*[1-(L/N) = T3*[N/N - L/N] = T3*[(N-L)/N]
N-L is an integer say M then
T2 = T3*(M/N) = M*(T3/N) = M*U
Thus we have BASIC RELATION Equations for T1 & T2 with T3 form as
T1 = L*U & T2 = M*U where T3 = N*U & U is an integer = T3/N
Also T3 = N*U implies a Skeletal Expression of T3 that have value of Unit Term U as an integer = T3/N. Similarly T1 = L*U & T2 = M*U implies that T3. T1 & T2 have at least one each Skeletal Expression that have Common Unit Term U = T3/N which further implies that for T3 = T1 + T2 its terms T3, T1 & T2 are Multiples of a Common Unit Term U of integer value as it is claimed by Lemma-1
[IIIC] Based on Lemma-1 EXPANSION for 3 integers expressed as T3 = T1 + T2
& Related RULES
By Basic Relation Equations
T1 + T2 = L*(T3/N) + M*(T3/N) = (L+M)*(T3/N) = T3 which implies L+M = N
Also U = T3/N gives T3 = N*U which implies that T3 = T1 + T2 have an Expansion
as T3 = N*U = (L+M)*U = L*U + M*U = T1 + T2
which implies the following RULES
(1) T3 = N*U, T1 = L*U & T2 = M*U implies Skeletal Expressions of T3, T1 & T2
where N = L+M implies that One of the Skeletal Expression of T3 is bifurcated to form the Skeletal Expressions of T1 & T2 where Value of Unit Term U is kept unchanged &
Number of Unit Terms N is bifurcated as N = L+M
(2) which implies that U = integer forms as Common Unit Term in T3, T1 & T2
where T3 = N*U, T1 = L*U & T2 = M*U
(3) For the cases where U> 1 for T3 = T1 + T2 its terms T3, T1 & T2 have Common Factor caused by Common Unit Term U
[IIID] Implications of Lemma-1 & its related RULES on Beal Conjecture
Beal Conjecture C^z = A^x + B^y belongs to Lemma-1 Equation T3 = T1 + T2
where T3 = C^z, T1 = A^x & T2 = B^y which implies that the Beal Conjecture keeps
the following Lemma-1 Rules & particularities
(1) Basic Relation Equation of A^x & B^y with C^z form as
A^x = L*(C^z/N) = L*U & B^y = M*(C^z/N) = M*U where U is an integer = C^z/N
such that L, M & N are integers where N = L+M
Ex: for C^z = 9^4 = 3^6 + 18^3
N = 9 = L+M = 1 + 8 gives A^x = 1*(9^4/9) = 3^6 & B^y = 8*(9^4/9) = 18^3
(2) Expansion of C^z = A^x + B^y for C^z = N*U forms as
C^z = N*U = (L+M)*U = L*U + M*U = T1 + T2 where proper values for L, M & U
L*U gives A^x & M*U gives B^y
Ex: C^z = 9^4 = N*U = 9*9^3 = (1+8)*9^3 = 3^6 + 18^3 = A^x + B^y
(3) As explained in [IIG] for Beal Conjecture Term C^z = U*N
U & N are integers > 1 & C^z = U*N = F1*F2 where F1 & F2 are 2 Factors in C^z &
For Lemma-1 based bifurcation of C^z to 2 integers such that C^z = T1 + T2
where T1 = L*(C^z/N) = L*U & T2 = M*(C^z/N) = M*U there is Common Factor in C^z, T1 & T2 caused by U > 1 & for proper values of L, M & U the terms T1 & T2
form as A^x & B^y
which implies that C^z, A^x & B^y have Common Factor caused by Common Unit term U > 1 &
U = F1 form as EQUAL sides for the Rectangles or Square Figures formed for C^z, A^x & B^y
which implies PROOF for Common Factor in Beal Conjecture Terms C^z, A^x & B^y
by methods in Geometry
Ex: For Beal Conjecture 72^4 = 12^6 + 288^3 where U = 12^6 & N = 9 for the value L+M = 1 + 8 = 9 = N
Rectangles for A^x = 12^6 = 1*12^6, B^y = 288^3 = 8*12^6 & C^z = 9*12^6 where
EQUAL SIDES for U = 12^6 PROVE COMMON FACTOR in C^z, A^x & B^y
[IIIE] PROOF for BEAL CONJECTURE
By [IIID] we have PROOF for Common Factor in C^z, A^x & B^y of Beal Conjecture
C^z = A^x + B^y & Common Factor in C^z, A^x & B^y cause Common Prime Factor
in A, B & C as explained below
C^z & C have same Prime Factors.
A^x & A have same Prime Factors and B^y & B have same Prime Factors
which implies that C, A & B have Common Prime Factor caused by Common Factor in C^z, A^x & B^y
Ex(1): For 2^13 = 4^6 + 16^3 Common Factor 2^12 in the terms for C^z, A^x & B^y cause Common Prime Factor 2 in C = 2. A = 4 & B = 16
Ex(2): For 72^4 = 12^6 + 288^3 Common Factor 12^6 in the terms for C^z, A^x & B^y cause Common Prime Factors 2 & 3 in C = 72, A = 12 & B = 288
Ex(3): For [2*(49)^2]^3 = 49^6 + 7^13 Common Factor 7^12 in the terms for C^z, A^x & B^y cause Common Prime Factor 7 in C = 2*(49)^2, A = 49 & B = 7
Therefore we PROVED that for Beal Conjecture C^z = A^x + B^y the Base Numbers A, B & C have Common Prime Factors &
the BEAL CONJECTURE is PROVED
[IIIF] CONCLUSION
By the following Sub PROOFS we have a PROOF for BEAL Conjecture
(1) By Lemma-1 explained in [III] we have A^x = L*U, B^y = M*U where C^z = N*U such that U is an integer = C^z/N where C^z is divisible by integer N
(2) By [II] Skeletal Expression of integers is explained & C^z = N*U, A^x = L*U & B^y = M*U imply Skeletal Expressions of C^z, A^x & B^y which further implies that
integer U = C^z/N which is Unit Term for Skeletal Expression of Beal Conjecture Term C^z forms as Common Unit Term to C^z, A^x & B^y
(3) By [II G & H] we verified that for Skeletal Expression of Beal Conjecture C^z value of U = 1 is invalid & for C^z = U*N value of U > 1 & U = C^z/N is a factor in C^z & taken as U = F1 &
Common Factor in C^z, A^x & B^y is PROVED where C^z = U*N, A^x = U*L & B^y = U*M
(4) By [IIIF] we explained that C, A & B have all the Prime Factors that are in the Common Factor in C^z, A^x & B^y & which implies that PROOF for Common Factor Directly implies that C, A & B have Common Prime Factor &
thus we have the PROOF for BEAL CONJECTURE
