[Part [ A-2 ] LEAST VALUES of IMPORTANT TERMS in BEAL CONJECTURE

[2A] LEAST value of C in C^z

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By conditions of Beal Conjecture A, B & C are
positive integers > 0. Which implies
A^x + B^y > 1^z 
Therefore for valid terms in Beal Conjecture C^z = A^x + B^y,
C must be integer > 1

i,e. C = integer 2 or more
Therefore C^z is a composite integer

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[2B] LEASET value of  VALID Term of  C^z, A & B 

in BEAL CONJECTURE C^z =  A^x + B^y

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We verified above in [2A]

for BEAL CONJECTURE C in C^z must be integer 2 or more,

Also

for Beal Conjecture  z must be +ve integer > 2 
Therefore for Beal conjecture value of C^z = 2^3 or more 
Which implies for Beal Conjecture
1^x + 1^y > C^z 

since 1^x + 1^y = 2 & C^z = 2^3 or more
 A & B can't be simultaneously = 1

which implies A & B are integers > 1 = 2 or more

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Also it is already proved that for,

Q^n = P^m + 1 where P,Q, m, & n are positive integers,

the only solution is 3^2 = 2^3 + 1 [Proved by Preda Mihailescu for Catalan Conjecture]

But equation  3^2 = 2^3 +1 is invalid equation for Beal Conjecture as x, y & z must be integers > 2

[2C ] Therefore for valid terms in Beal Conjecture C^z = A^x + B^y,

C, A & B are integers > 1 = 2 or more.

Which implies in BEAL Conjecture C^z = A^x + B^y

C^z, A^x & B^y are COMPOSITE INTEGERS

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Least Value for C^z = 2^3 = 8

Similarly,

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

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

where value of Unit Term  U = C^2 & Number of Unit terms N = C^(z-2)

Least Value for C^2 = 4 &

Least Value for C^(z-2) = 2^(3-2) = 2

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PART 1 of ROOF for BEAL CONJECTURE

[I] BEAL CONJECTURE states that

if C^z = A^x + B^y

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

x, y & z are integers > 2

then A, B & C have COMMON PRIME FACTOR

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

[IB] EXAMPLES for BEAL CONJECTURE C^z = A^x + B^y

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

(1) 2^13 = 4^6 + 16^3 .... [Common prime factor 2 ]

(2) 9^4 = 3^6 + 18^3 .... .. [Common prime factor 3

(3) (2*49^2)^3 = 49^6 + 7^13 .... [Common prime factor 7]

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

[IC] By the following method or similar methods

several VALID terms for BEAL CONJECTURE can be generated.

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

By FLT 

For x*y = n where x, y & n are integers > 2

A^n + B^n = K where K can't be expressed as K = C^n

Then

K^n*(A^n + B^n) = K*K^n &

[(K*A)^y]^x + [(K*B)^x]^y = K^[(x*y)+1]

gives Terms for A^x + B^y = C^z

where K cause Common Prime Factor in A, B & C

Ex: n = x*y = 3*4 = 12  where A = 2 & B = 3

Then

(2^3)^4 + (3^4)^3 = K & multiplying by K^3*4 = K^12

[(K*2)^3]^4 + [(K*3)^4]^3 = K^12+1

forms as terms for Beal Conjecture C^z = A^x + B^y

where K cause Common prime Factor in A, B & C

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