🚀 go-pugleaf

In article <1992Nov2.090701.1@tnclus.tele.nokia.fi>,
hporopudas@tnclus.tele.nokia.fi writes:

 ...deleted part...
>
>
>   Could any of you say where the error in the following reasonig ?:
>                        -----------------------------------------
>
>   Combining (8') and (7) we have also
>
>               n   2          n   n 2  (1/2)      1     n     n
>            ((x -1) + (1/4)( y - x )  )     > ( ------ )  - (x -1)
>                                                 1/n
>                                                2   -1
>   From
>             n   n          1      2n       n         1    n  (1/2)
>            y - x  > 2( ( ------- )   - 2( x - 1) ( ----- )  )
>                            1/n                      1/n
>                           2   -1                   2   -1
>

 This last inequality is questinable ,because by suitable rearragements of
 terms you get also

              n   n          1      2n       n         1    n  (1/2)
             y - x  < 2( ( ------- )   - 2( x - 1) ( ----- )  )
                             1/n                      1/n
                            2   -1                   2   -1



>
>   This last inequality and the result of prof. Inkeri
>   (finnish mathematician), which states that
>
>                 3n-4
>          x  >  n     ; for n > 3
>                              =
>   (limitations?)
>


    Corrected form of this K. Inkeri's result is

                        3n-4
           x  >  (1/2) n     ; for n is prime exponet > 2

    (0 < x < y < z, x,y,z are relatively-prime integers)


    Two cases of Fermat's last theorem:
    ----------------------------------
                                         3
                                       2n + n   n
    If (xyz,n)=1, (Case I), then x > ( -------- )  , n is prime exponent > 2.
                                       log(3n)

    (Case II is when n divides z, n is prime exponent > 2).




  Reference:

  Inkeri, K, 1953.
  Absch{tzungen fur eventuelle L|sungen der Gleichung
  im Fermatschen Problem.
  Ann. Univ. Turku, Ser. A, vol. 16, no. 1, pages 3-9.



>   We have achieved the following impossible inequality
>
>          3n-4                  1      n    (1/n)
>         n     < x <  ((1/2)( ------ )  + 1)
>                   =           1/n
>                              2   -1
>
>   because left side of the the inequality grows always faster than right
>   side of the inequality, when n is greater than 2.
>


 I must reinvestigate what is followed from combining (8') and (7).
 Above impossible inequality is under adjustment at the moment. It
 is possibly wrong.


 ---Hannu.