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ATTENTION !  May be best viewed by  INTERNET EXPLORER 5.0 or HIGHER

THEOREMS  OF  EL'ALIM  STATES  THAT ;

PART-1

If   Codd   is an odd  composite    (e.g.  9 , 15 , 21 , 25 , 27.........)   then  ;
there  is   always   an   integer  solution  to  the  diophantine's   equation ,

       (R and  n , belonging  to  the  finite set  of  integers)

The two divisors of   Codd  in that case will  be ;

           such that   Codd  =d1.d2

if  Codd  is  an  odd  Prime  P  (e.g. 3 , 5 , 7 , 11)   then  we  may  never  be able to  find
an integer solution to the diophantine's equation;  

     Any   odd   composite  number   Codd9   can  be  expressed  as ;   

      where  R2  and   n2   are   certain integers.

      On the other hand , no  odd  Prime Number  P  can be  expressed as ;    

This  theorem  provides a new definition of distinction between all  odd  composites , and   odd  primes .

Some Illustrative Examples;

Codd =5.13=65           Odd  Composite
R=9 ,  n=14               these values provide  the solution to diophantine's equation

Codd =7.11=77           Odd  Composite
R=9 ,  n=18               these values provide  the solution to diophantine's equation

Codd =13.17=221       Odd  Composite
R=15 ,  n=54             these values provide  the solution to diophantine's equation

 Codd =101.7=707       Odd  Composite
R=54 ,  n=153           these values provide  the solution to diophantine's equation

Codd =5.103=515       Odd  Composite
R=54 ,  n=104           these values provide  the solution to diophantine's equation

P =103                   Prime
R , n ;                         no  value  provide  the solution to diophantine's equation

 P =37                     Prime
R , n ;                         no  value  provide  the solution to diophantine's equation

PART-2

If            where;

P = Any odd prime  such that    ,  p ³ 3
and ,
x , n = Any natural numbers   such that   n ³ 1    and     n ¹ (P - 1)/2

then, the quadratic eqn.  doesn't  possess  any  integer  roots   (x1 and x2 )

Part-2  and  Part-1  may be extracted from each other.

PART-3

If           where;

(m > t    and    m1)   and   ( t1 )   and  (x)  ,  are natural numbers

such that  ,     is  an  integer  , 

then , the quadratic eqn.  doesn't  possess  any  integer  roots   (x1 and x2 )
Part-3   may  be  extracted   from   a  cubic  Fermat  aquation .

PART-4

The impossible triple product

If   a , b , L  are  any  integers,  then there is no solution
to the following Diophantine's  relation , except the trivial one.  (L = 0 , a = - b )

3.(a+b).(L-a).(L-b) = L3

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