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ATTENTION !  May be best viewed by INTERNET EXPLORER 5.0 or HIGHER
THEOREM OF EL'AZIZ  STATES THAT ;

suppose we have a cubic equation as defined;
 a3 + b3 + c3 = k3      such as  ,     { 33 + 43 + 53 = 63      or    333 + 163 - 343 = 93   }  .

Then we may find functions   F1(a,b,c,k)     ,    F2(a,b,c,k)    ,   F3(a,b,c,k)     ,   F4(a,b,c,k)
such that  ;                              F1(a,b,c,k) 3  +   F2(a,b,c,k)3  +  F3(a,b,c,k)3   =  F4(a,b,c,k)3

                Philosophical importance of such a theorem is that, some equations grow from similar  kinds  of  equations  which  are  used  as  seed  . This  is  like , growing of living
organisms from a seed of similar origin.


Suppose we have the equation    -->      a3 + b3 + c3 = k3   
Then We'll define the integer variables as;

d = a + b + c - k                                   ,      P = [d + 2.(k - c)].[6.(a + b).(k - c).(k - a) - d3 ]
 
R = d.[6.(a + b).(k - c).(k - a) - d3 ]       ,        A = (b2 - a2 )  +  (k2 - c2 )

The  first  set of cubics will be

F1(a,b,c,k) = P  + a.d3                     
F2(a,b,c,k) = R + b.d3        
F3(a,b,c,k) = R + c.d3
F4(a,b,c,k) = P  + k.d3

F1(a,b,c,k)3 + F2(a,b,c,k)3 + F3(a,b,c,k)3  =  F4(a,b,c,k)3

The  second  set of cubics will be

F1/(a,b,c,k) = P  + a.d3 + A.d2
F2/(a,b,c,k) = R + b.d3  - A.d2
F3/(a,b,c,k) = R + c.d3 + A.d2
F4(/a,b,c,k) = P + k.d3 + A.d2

F1/(a,b,c,k)3+ F2/(a,b,c,k)3 + F3/(a,b,c,k)3  =  F4/(a,b,c,k)3

Please also observe the beauty of the fact that , if  -->   a3 + b3 + c3 = k3
then ;      12.(k-a).(k-c).(a+b).(b+c) = d4 + 3.A2


Please click here to see the related Mathematica program

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