Best solutions to - "Numbers of the kind 0.aaa…=1aaa…a"

Deven Livingston

Answered question

2022-04-23

Numbers of the kind

0 . a a a \dots = \frac{1}{a a a \dots a}

Answer & Explanation

gonzakunti2

Beginner2022-04-24Added 16 answers

Step 1
I think a complete answer should cover all bases (no pun intended). We will have:
$a = \frac{b^{d} - 1}{\sqrt{b^{k d} - 1}}$
for any base b. However for $k \geq 2$ :
$\sqrt{b^{k d} - 1} \geq \sqrt{b^{2 d} - 1} = \sqrt{(b^{d} - 1) (b^{d} + 1)} \geq b^{d} - 1$
forcing $a < 1$ . Hence we must have $k = 1$ , with
$a = \sqrt{b^{d} - 1}$ or $b^{d} - a^{2} = 1$
By Catalan's conjecture (or Mihăilescu's theorem, or if you are interested, this $1 + a^{2} = b^{d}$ is a special case proven by V. A. Lebesgue using Gaussian integers), if $d \geq 2$ there are no solutions. This forces $d = 1$ , and we have only the uninteresting case:
$1 + a^{2} = b$
so our choice of b must be one more than some square. For base 10,
$10 = 1 + 3^{2}$ , so $0 . \dot{3} = \frac{1}{3}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Pre-Algebra

Didn't find what you were looking for?