🚀 go-pugleaf

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