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

THE  THEOREMS  OF  EL'AHIR  STATES  THAT ;

Binomial coefficient  C(x,y)  is given by ---    


PART-1
If    n 1 , a , b   are integers  such that;   k=a/b    ,  and an angle    defined as ;

        ,  Then      is  a  rational number  , that is ;
 

       is  a   rational number.

     is  an  integer.

as a proof  , we may state the following relation.
 


Since the right hand side of the equation is a rational    ,     is  also
a  rational.

Please click here to see the related Mathematica program


PART-2
If    n 1 , a , b   are integers  such that;   k=a/b    ,  and an angle    defined as ;
 

ATTENTION  n Being odd   (n = 1,3,5......)


    ,  Then      is  a  rational number  , that is ;
 

       is  a  rational  number.

      is  an  integer.

  Provided that  n  is an odd integer .

as a proof  , we may state the following relation.

 


Since the right hand side of the equation is a rational   ,       is  also
a  rational  for  odd  n .

Please click here to see the related Mathematica program


PART-3
If    n 1 , a , b   are integers  such that;     ,  and  an  argument 
defined as ;
 
      Then     is  a  rational  number   that is ;

     is  a  rational  number.

      is  an  integer.

as a proof  , we may state the following relation.


 


Since the right hand side of the equation is a rational   ,    is  also
a  rational.

Please click  here to see the related Mathematica program

PART-4

If    c   is an  odd  integer 1   , then for the following equation ;

 t  is  also  an  exact  odd  integer.

As a proof  , we may state the following relation  for odd  c ;



Right  side  is  an  exact  odd  integer  for  odd   c 1  so  the  left  side  either.
Please click  here to see the related Mathematica program. 

PART-5

The nested transcendental chain of functions (Cos , Sin) ;

If (m) is any integer greater than one ,    (e.g   2 , 3 , 4 , 5 , 6 ............ )

And       n1 , n2 , n3 ..................nj     are  integers  of  the  type   (4*k+1) ,    (e.g   5 , 9 , 13.....)

Then we have the following relations holding true;

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Nested transcendental chain of the  first  order (Cos , Sin) ;

f1(n,m) = Cos[n1 . ArcCos[m]]

f2(n,m) = Sin[n1 . ArcSin[m]]

Then     f1(n,m) =  f2(n,m)   ,     (the expressions being equal to an integer )

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Nested transcendental chain of the  second  order (Cos , Sin) ;

f1(n,m) = Cos[n2 . ArcCos[Cos[n1 . ArcCos[m]]]]

f2(n,m) = Sin[n2 . ArcSin[Sin[n1 . ArcSin[m]]]]

Then     f1(n,m) =  f2(n,m)   ,     (the expressions being equal to an integer )

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Nested transcendental chain of the  third  order (Cos , Sin) ;

f1(n,m) = Cos[n3 . ArcCos[Cos[n2 . ArcCos[Cos[n1 . ArcCos[m]]]]]]

f2(n,m) = Sin[n3 . ArcSin[Sin[n2 . ArcSin[Sin[n1 . ArcSin[m]]]]]]

Then     f1(n,m) =  f2(n,m)   ,     (the expressions being equal to an integer )

Please click  here to see the related Mathematica program. 

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Nested transcendental chain of the   j'th   order (Cos , Sin) ;

f1(n,m) = Cos[nj .................................[n1 . ArcCos[m]]]]]...]

f2(n,m) = Sin[nj ................................. [n1 . ArcSin[m]]]]]....]

Then     f1(n,m) =  f2(n,m)   ,     (the expressions being equal to an integer )

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