[3-a] SIMPLIFIED STEP (1)
A PROOF for COMMON FACTOR in any of the 2 terms in
BEAL CONJECTURE C^z = A^x + B^y
is sufficient to PROVE BEAL CONJECTURE as follows:
...............
Let C^z & A^x have COMMON FACTOR = D
which implies that C^z = K2*D & A^x = K1*D where K1 & K2 are integers such that K2 > K1 (since C^z > A^z)
C^z = A^x + B^y gives
B^y = C^z - A^x = K2*D - K1*D = (K2 - K1)*D
(K2 - K1) is an integer which implies that C^z, A^x & B^y have COMMON FACTOR
...........
[3-a] SIMPLIFIED STEP (2)
C^z & C have the same PRIME FACTORS
Similarly A^x & A have the same PRIME FACTORS
Also B^y & B have the same PRIME FACTORS
which implies that
a PROOF for COMMON FACTOR in C^z, A^x & B^y
automatically PROVES that C, A & B have COMMON PRIME FACTOR
that are in the COMMON FACTOR in C^z, A^x & B^y
...........
[3-b] SIMPLIFIED STEP to PROVE BEAL CONJECTURE
Based on simplified STEPS [3-a] (1) & (2)
.............
in this Article
A proof for COMMON FACTOR in Beal Conjecture Terms C^z & A^x
is given to PROVE BEAL CONJECTURE as follows,
...................
(1) A PROOF for COMMON FACTOR in C^z & A^x automatically
PROVES that C^z, A^x & B^y have COMMON FACTOR.
..........
(2) C & C^z have SAME PRIME FACTORS,
A & A^x have SAME PRIME FACTORS &
B & B^y have SAME PRIME FACTORS
Therefore
A proof for COMMON FACTOR in Beal Conjecture Terms C^z & A^x
proves that C^z, A^x & B^y have COMMON FACTOR
which implies that
in BEAL CONJECTURE C^z = A^x + B^y
C, A & B have COMMON PRIME FACTORS that are in
the COMMON FACTOR in C^z, A^x & B^y
which is the PROOF FOR BEAL CONJECTURE
