Beal conjecture - 4

Beal conjecture-4

In the three members like or unlike power relations both the Fermat’s assertion and Beal conjecture have some similarities and differences.

Similarities

1. All the members can be even, then the relation will be reducible.. All the member cannot be odd due to non-conservation of oddness.

2. One or two members but not all the three may be square or higher power but all the members cannot be square or with the same higher power.

3. FLT and Beal conjecture are mathematically true with fractional and rational or complex numbers.

4 . In irreducible form, FLT  and the Beal conjecture do not have a common prime factor among the members.   

Differences

1.According to FLT in ax  + by  = cz , a,b,c the base numbers cannot be positive integers when the exponents x,y,z are same and greater than 2.

 (an – 1)n  + (an  - 1)n+1   = [a(an  - 1)]n , where a and n are any two independent variables which gives a set of multi-power relations obeying Beal conjecture with exponents  (n,n+1,n).

[a(an + bn)]n  + [b(an  + bn)]n    = (an + bn) n+1  gives a set of multi-power relations with exponents (n,n,n+1)

[2n an+1]n+2  +  [2n an+1]n+2   = [2n+1 a n+2] n+1  gives  a set of such relatins with exponents  (n+1,n+1,n) 

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}

cubical relations

One cube is equal to sum of three cubes

According to Fermat, a³  +  b³  =  c³   is not possible with a,b,c are all positive integers. However, one cube can be expressed as a sum of three cubes.

6³  = 5³  + 4³  + 3³

9³   = 8³  + 6³  +  1³

20³  = 17³  + 14³ + 7³

25³  =  22³ +  17³  + 4³

28³  = 21³  + 19³  + 18³

58³  = 49³  + 42³ + 15³

70³ = 57³  + 54³ + 7³

210³  = 171³  + 162³  + 21³

214³  =  213³ + 51³  + 16³

256³  = 255³  + 57³  + 22³

256³  = 255³  + 27³  + 22³

298³  = 297³ + 64³  + 15³

682³  = 675³  + 213³  + 16³

1062³  = 1059³  + 213³  + 108³

1408³  = 741³  + 675³  + 346³

2176³ = 2169³  + 447³  + 214³

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.