Wednesday, October 11, 2017

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) 

Tuesday, October 10, 2017

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  + b)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 +   2 =  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  = 2m where a and b  are equal to some power of 2 , since sum of  (2m)n + (2m)is always expressible as a power of 2 and is equal to 2mn + 1          

                                               2n  +   2n    =  2n + 1
                                                      4n    +  4n    =  22n + 1
                                                     8n    +  8n   =  23n + 1
                                                    16n  +  16= 24n + 1

                                                   32 + 32 =  25n + 1

cubical relations

One cube is equal to sum of three cubes
According to Fermat, a3  +  b=  c 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
20= 173  + 143 + 73
253  =  223 +  173  + 43
283  = 213  + 19+ 183
583  = 493  + 423 + 153
703 = 57+ 543 + 73
2103  = 1713  + 1623  + 213
2143  =  2133 + 513  + 163
2563  = 2553  + 573  + 223
2563  = 255+ 273  + 223
298= 2973 + 643  + 153
682= 6753  + 2133  + 163
10623  = 1059+ 2133  + 1083
14083  = 741+ 6753  + 3463

21763 = 21693  + 4473  + 2143

Sunday, October 8, 2017

Beal Conjecture -2

Irreducible form of  three member multi-power relation (Beal conjecture)
According to Beal all three member multi-power relation a+  by   = c, 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 2 à     22  +  25  = 62
   X 33   à     33  +  62  =  35
  X 43   à    4+  8 =  242  
 X 36    à   36  +  183  =  38   (  X 93  à  93  +  183  = 812)
X 46  à 4+  32 = 1922 (  x 163  -à 163  + 215  = 1922 )
One of the properties of the multi-power relation in the form a +  bn  = c is existence of common multiplier  knm
a +  bn  = cm   x   k mn   -à  (km a)n  + (km b)n  =  (kn c)m
 There are many irreducible form of multi-power relation. One of the general forms is
[ 1 + nm  = ( nm  + 1) ] x  (nm  + 1)m  -à (nm  + 1)m  + [n(nm  + 1)m ] = ( nm  + 1) m+1

Where m and n may have all possible values.