Math Forum / Mathematics / Undergraduate Math / September 2005

Hi,

  Let p be a prime number. I need to show that

             2n+1
            p     +   1
       K = ---------------
              p + 1

where n >= 0 is an integer. This seems to work fine with examples

            (i.e. (5^3 + 1)/(5+1) = 21)

but I'm not sure how to show this. Induction on n:

Case 0:
           0 + 1
          p       +   1
        ----------------- = 1
          p + 1

inductive step:

  suppose it is true for k,

         2k+1
        p     +   1
     ---------------
        p + 1

we need to show it is true for k + 1:

      2(k + 1) + 1                2k + 3
     p              +   1        p       + 1
  ------------------------   = ------------------
        p + 1                       p + 1

but I don't see how to work the induction hypothesis in there.

The formula seems reminiscent of the Fundamental Theorem of Algebra
but I don't see how to use it.

Thank you,
-e