By Part A-2
We have verified that in Beal Conjecture C^z = A^x + B^y
C, A, & B are integers > 1 &
C^z, A^x & B^y are COMPOSITE integers
..........
[1] By Rules in Geometry
If C^z & A^x are represented by Rectangle or Square Figures
with sides in integers value
then
Equal sides that have value > 1 projects the FACTORS
that are common in C^z & A^x &
prove COMMON FACTOR in C^z & A^x
which implies that
For Rectangles formed for C^z = 1*C^z & A^x = 1*A^x
Equal sides represent integer 1 & sides that represent C^z & A^x
are UNEQUAL
Also sides that represent C^z & A^x represent 2 terms that are multiple of 2 or more FACTORS
which implies that
Rectangles formed for C^z = 1*C^z & A^x = 1*A^x
are meaningless to PROVE or DISPROVE COMMON FACTOR in C^z & A^x
.........
C^z/N = U where N is integer & C^z is divisible by N gives
C^z = U*N & Rectangle or Square figures for C^z
that have sides in integer value
............
For the value N =1 & N = C^z itself
C^z/N = U gives C^z = U*N = 1*C^z or C^z = U*N = C^z*1
which implies that
For the values N =1 & N = C^z in C^z/N = U
Rectangles formed for C^z = U*N
have sides that represent integer 1 or C^z itself &
that Rectangles are meaningless
to PROVE COMMON FACTOR in C^z & A^x
.........
For C^z/N = U where N is an integer > 1 & N < C^z
U is an integer > 1
which implies that
for proper values of N every FACTOR in C^z can be projected as U
such that C^z = U*N = F1*F2 where F1 & F2 are 2 factors in C^z
...........
Also by RULES in Mathematics
If a COMMON FACTOR is verified in 2 INTEGERS T1 & T2
by 2 Rectangle or Square Figures for that 2 integers or
any other methods
that VERIFIED COMMON FACTOR in T1 & T2
is VALID for ever & for all cases
including the Rectangles formed for T1 = 1*T1 & T2 = 1*T2
...........
Ex: Case(1): Let T1 = 15 & T2 = 18
In the case of Rectangles formed for 15 = 3*5 & for 18 = 3*8
Side = 3 form as Equal sides for T1 = 15 & T2 = 18 & PROVES that
T1 = 15 & T2 = 18 have COMMON FACTOR 3
.........
Ex:Case(2)
In the case of Rectangles formed for 15 = 1*15 & 18 = 1*18
Equal sides represent integer 1 &
there isn't any PROOF or DISPROOF for
COMMON FACTOR in T1 = 15 & T2 = 19
But COMMON FACTOR verified for T1 & T2 by Case(1) is VALID
for the Case(2)
where Rectangles formed for T1 = 1*15 & T2 = 1*18
...............
Which implies that
For the values N > 1 & N < C^z
C^z/N = U gives C^z = U*N = F1*F2 where F1 & F2 are 2 Factors
in C^z &
Rectangles or Square Figures formed for
C^z = U*N = F1*F2
are SUFFICIENT to verify COMMON FACTOR C^z & A^x
by COMPARING it with Rectangle or Square figures formed for A^x
& thus to PROVE that whether
C^z & A^x have COMMON FACTOR or NOT
..................
