Beal conjecture-3

Beal conjecture-3 : Methods  of getting  three member  multi-power relations

Method-1 

It is suitable to obtain multi-power  relation  of type  aⁿ  + bⁿ  = c ⁿ⁺¹

The sum of any two like powers when raised to same power will be a common multiplier.

 aⁿ  + bⁿ  =  (aⁿ  + bⁿ ) 

Multiplying with the common multiplier (aⁿ  + bⁿ )ⁿ , we get [a (aⁿ  + bⁿ )]ⁿ  + [b (aⁿ  + bⁿ )]ⁿ  =

(aⁿ  + bⁿ  )ⁿ⁺¹  , where a and b are any positive integers.

    a       b       n      irreducible relation                  multiplier         multi-power relation

    1       2       3           1 +   2³   =  9                                  9³                         9³ + 18³  =  9⁴

    1       2        4           1 +   2⁴   =  17                              17⁴                       17⁴ + 34⁴  = 17⁵

      1       2       5           1  +  2⁵   =  33                              33⁵                          33⁵  + 66⁵  = 33⁶

    1      3        3           1  + 3³   = 28                                28³                           28³  + 84³  =  28⁴

     2      3        3           2³  + 3³   = 35                               35³                        70³  + 105³  = 35⁴

     3       3        3           3³   + 3³ =  54                              54³                        162³  + 162³  = 54⁴

     2       3        4           2⁴  +  3⁴  = 97                              97⁴                    194⁴ +  194⁴   =  97⁵

Method-2

It is suitable to get multi-power relation of type  aⁿ  + aⁿ  = 2^m where a and b  are equal to some power of 2 , since sum of  (2^m)ⁿ + (2^m)ⁿ  is always expressible as a power of 2 and is equal to 2^{mn + 1}          

                                               2ⁿ  +   2ⁿ    =  2^{n + 1}

                                                      4ⁿ    +  4ⁿ    =  2^{2n + 1}

                                                     8ⁿ    +  8ⁿ   =  2^{3n + 1}

                                                    16ⁿ  +  16ⁿ  = 2^{4n + 1}

                                                   32ⁿ   + 32ⁿ   =  2^{5n + 1}