🚀 go-pugleaf

In article <1992Oct29.093106.1@tnclus.tele.nokia.fi>,
hporopudas@tnclus.tele.nokia.fi writes:
>
> In article <1992Oct28.161726.1@tnclus.tele.nokia.fi>,
> hporopudas@tnclus.tele.nokia.fi writes:
>>
>> Hello !
>>
>> Does any of you know inequalities related to z in Fermat's last theorem.?
>>    n   n    n
>> ( x + y  = z ; x,y,z,n positive integers, x < y, n > 2 )
>>
>> ---Hannu.
>>
>
>  I have discovered some years ago following inequalities for x, y, z and n
>  (int(v)=integer part of v, log is natural logarithm):
>
>  -----------------------------------------------------------------------
> !                              log2                                    !
> !              3 < n < int( ---------- )                               ! (1)
> !                =   =            z                                    !
> !                           log(-----)                                 !
> !                               z - 1                                  !
>  -----------------------------------------------------------------------
> !                             z            z                           !
> !              1 < x < int( ------- ) <  -------                       ! (2)
> !                =   =         1/n    =     1/n                        !
> !                             2            2                           !
>  -----------------------------------------------------------------------
> !               z            z                                         !
> !              ----  < int( ----- ) + 1 < y < (z - 1)                  ! (3)
> !                1/n          1/n       =   =                          !
> !               2            2                                         !
>  -----------------------------------------------------------------------
> !              n      n 1/n                   n        z       n 1/n   !
> !   z-int(z -(z -(z-1) )   ) < x < z-int(z -(z -(int(------)+1) )   )  ! (4)
> !                            =   =                      1/n            !
> !                                                      2               !
> ------------------------------------------------------------------------
>
>
>  Also the following inequalities holds:
>
>                              n
>           n       n    n    z
>          z - (z-1)  < x <  ---                                  (5)
>                             2
>           2
>          x            xy
>        --------  <  --------  < z                               (6)
>          n   1/n      n   1/n
>        (x -1)       (x -1)
>
>              1/n
>             2
>           ------- < z                                           (7)
>            1/n
>           2   -1
>                   n                       n
>          1     -(x - 1)         1        x - 1   2    1   1/2
>         ---- >  --------- +  ------- ( (--------)  + --- )      (8)
>           n       n   n 2      n   n      n   n       2
>          z      (y - x )     (y - x )    y - x
>
>
>
>  Fermat's equation has also the following alternative series form:
>
>     ----------------------------------------------------------
>    !            n             2n                    3n        !
>    !  z - x    y   -n  (n-1) y    -2n  (n-1)(2n-1) y   -3n    !
>    !  ----- = --- z  + --------- z   + -------------- z   +...!  (9)
>    !    z      n             2                3               !
>    !                     2! n             3! n                !
>     ----------------------------------------------------------
>    !   n                                                      !
>    !  y              k1                                       !
>    ! --- = constant z  + ....   ,  0 < constant < (z-1)       !  (10)
>    !  n                              =          =             !
>    !                          n                               !
>    !                         y                                !
>    ! where  k1 = int( log ( ---- )),      (base-z logarithm)  !
>    !                     z   n                                !
>    !                                                          !
>     ----------------------------------------------------------
>
>
>
>  ---Hannu

 As a matter of fact this constant > 0.   (k1 = k_1)

 Similarly

              2n
       (n-1) y                k_2
       --------- = constant  z   + ... , 0< constant   < (z-1)
             2             2                        2  =
         2! n

       where
                                   2n
                            (n-1) y
         k_2 = int ( log (  --------- ))
                        z         2
                              2! n

        ...    ...   ...     ...    ...


                                    in
        (n-1)(2n-1)... ((i-1)n -1) y                k_i
        -----------------------------  = constant  z   + ...  ,
                       i                         i
                   i! n


        0 < constant   <  (z-1)  and
                    i  =

                                                     in
                         (n-1)(2n-1)... ((i-1)n -1) y
        k_i = int( log ( ----------------------------- ))
                      z                 i
                                    i! n


 (Idea is to try to prove that (9)'s right side is unperiodic number
  in the base system z.).


 ---Hannu.

 P.S. My employer denied me to write anything in sci.physics, sci.astro,
 sci.math, sci.philosophy.tech and talk.religion.misc because some from
 sci.physics or sci.math contacted to my employer due to my posting
 'Structure of Time' (which article was spoked in the name of God by my
 5 years old (28.4.1992) daughter Hanna-Maria).