Beal Conjecture -2

Irreducible form of  three member multi-power relation (Beal conjecture)

According to Beal all three member multi-power relation a^x  +  b^y   = c^z  , where a,b,c and the exponents x,y,z  are all positive integers such that x,y,z > 2, the base numbers a,b,c  will invariably have one more common factors.

If  all the members are expressed in the same power, any number can be used as common multiplier. But if the powers are different , only certain selective common multipliers will make reducible form of multi-power relations. If one of the members in the irreducible form of a multi-power relation is 1, it can be multiplied with any multiplier of any form..It is exemplified with a typical example.

1+ 8 = 9   =   1 + 2³  =  3²

    X 2²   à     2²  +  2⁵  = 6²

   X 3³   à     3³  +  6²  =  3⁵

  X 4³   à    4³  +  8³   =  24²  

 X 3⁶    à   3^{6  +}  18³  =  3⁸   (  X 9³  à  9³  +  18³  = 81²)

X 4⁶  à 4⁶  +  32³   = 192² (  x 16³  -à 16³  + 2¹⁵  = 192² )

One of the properties of the multi-power relation in the form aⁿ   +  bⁿ  = c^m   is existence of common multiplier  k^nm

aⁿ   +  bⁿ  = c^m   x   k ^mn   -à  (k^m a)ⁿ  + (k^m b)ⁿ  =  (kⁿ c)^m

 There are many irreducible form of multi-power relation. One of the general forms is

[ 1 + n^m  = ( n^m  + 1) ] x  (n^m  + 1)^m  -à (n^m  + 1)^m  + [n(n^m  + 1)^m ]^m   = ( n^m  + 1) ^m+1

Where m and n may have all possible values.