From the fact that 1 5 ( 3 + 4 i ) has infinite order in ( C , ⋅ ), I'm supposed to infer that 1 π arctan ⁡ 4 3 is irrational. Proving a arctan ⁡ 4 3 is irrational.

Question

From the fact that       1    5    (  3  +  4  i  ) has infinite order in   (      C    ,  ⋅  ), I&#039;m supposed to infer that       1    π    arctan  ⁡            4      3       is irrational. Proving a   arctan  ⁡      4    3   is irrational.

humanistex3 · Accepted Answer

Let   α be your complex number, then      α    n    =            a      n              5              n        +        1              +            b      n              5              n        +        1              iand since we have      α    n    =      1    5    (  3  +  4  i  )      (                  a                  n          −          1                            5        n              +    i                  b                  n          −          1                            5        n              )    =      1          5              n        +        1                  (    3          a              n        −        1              −    4          b              n        −        1              +    i    (          a              n        −        1              +    3          b              n        −        1              )    )  it is clear that a      a    n    =  3      a          n      −      1        −  4      b          n      −      1       and       b    n    =      a          n      −      1        +  3      b          n      −      1      , with       a    1    =  15 and       b    1    =  20.Examining everything     (  mod    7  ) we find   (      a    1    ,      b    1    )  =  (  1  ,  6  )  ,  (      a    2    ,      b    2    )  =  (  0  ,  5  )  ,  (      a    3    ,      b    3    )  =  (  6  ,  1  ) and   (      a    4    ,      b    4    )  =  (  0  ,  2  )  ,  (      a    5    ,      b    5    )  =  (  1  ,  6  ) so the cycle continues with no       b    n    =  0.Hence       α    n   is never real, but if   arctan  ⁡  (  4      /    3  )  =      p    q    π we would have   exp  ⁡      (          p      q        π    i    ⋅          2      q        )    =  1  =      α          2      q      .

From the fact that 1 5 </mfrac> ( 3 + 4 i ) has infinite order

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