Does there exist an irrational number x &gt; 2 such that any positive integer n can be written in the form n = a 0 + a 1 x + a 2 x 2 + …, where a i ∈ { 0 , 1 , … , 6 }?

Question

Does there exist an irrational number   x  &amp;gt;  2 such that any positive integer   n can be written in the form   n  =      a    0    +      a    1    x  +      a    2        x    2    +  …,  where       a    i    ∈  {  0  ,  1  ,  …  ,  6  }?

hildiadau0o · Accepted Answer

Let   x  =      7  . Any positive integer   n has a base-  7 representation, which is to say it can be written in the form  n  =      b    0    +      b    1    ⋅      7    1    +      b    2    ⋅      7    2    +  …where each       b    k    ∈  {  0  ,  1  ,  …  ,  6  }.Now, take       a          2      k        =      b    k   and       a          2      k      +      1        =  0, and you have the desired representation of   n.

Does there exist an irrational number x > 2 such that any positive integer n ca

Answered question

Answer & Explanation

New Questions in Algebra I