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THEOREMS OF EL'ADL STATES THAT ;
PART-1
If (m
> 1) is any integer ,
and p is any odd prime.
Then the following relations hold true ;
Cos[p.ArcCos(m)] º m Mod p
Cosh[p.ArcCosh(m)] º m Mod p
Cos[p.ArcCos(m)] =Cosh[p.ArcCosh(m)]
where (m,p) = 1
For a proof of
the fact that , left hand sides of the relations are
integers ;
please click here
.This theorem sieves off
many of the CARMICHAEL NUMBERS.
As a matter
of fact, perhaps these kind of theorems contain
less CARMICHAEL type of numbers,
when you
encounter with a pseudoprime to a certain m ; just change it
. Then the congruence may
hold true . Now
that means there are less pseudoprime
to all bases .
Please click here to see
the related Mathematica program
PART-2
are the
perpendicular legs of a right triangle.
and p is any odd prime ;
Then we'll have the following relation
;
where (k,p) = 1
For
a proof of the fact that , left hand
side of the relation is an exact integer ;
please click here
. This theorem sieves
off many of
the CARMICHAEL NUMBERS.
As a matter of fact, perhaps these kind of
theorems does't contain the concept of CARMICHAEL type of numbers, when
you encounter with a pseudoprime to a certain k , m ; just
change one or both of them .
Then the congruence holds true . Now that means
there is no pseudoprime to all
bases concept.
Please click here to see
the related Mathematica program
PART-3
Sin[p.ArcSin(m)] º (-1)( p-1) / 2 . m Mod p
where (m,p) = 1
For a proof of the fact that , left hand sides of the
relations are integers ;
please click here
.This theorem sieves off
many of the CARMICHAEL NUMBERS.
As a matter
of fact, perhaps these kind of theorems contain
less CARMICHAEL type of numbers,
when you
encounter with a pseudoprime to a certain m ; just change it
. Then the congruence may
hold true . Now
that means there are less pseudoprime
to all bases .
Please click here to
see the related Mathematica program
PART-4
If (m
> 1) is any integer ,
and p is any odd prime.
Then the following relations hold true ;
Sinh[p.ArcSinh(m)] º m Mod p
where (m,p) = 1
For a proof of the fact that , left hand sides of the
relations are integers ;
please click here
.This theorem sieves off
many of the CARMICHAEL NUMBERS.
As a matter
of fact, perhaps these kind of theorems contain
less CARMICHAEL type of numbers,
when you
encounter with a pseudoprime to a certain m ; just change it
. Then the congruence may
hold true . Now
that means there are less pseudoprime
to all bases .
Please click here to
see the related Mathematica program