# Exact Mandelbrot

# 1 Claim

The Mandelbrot set cannot be calculated with exact rational arithmetic, because the precision required goes up by a factor of two each iteration, so by as few as $100$ iterations, the storage space of the computer has long been exhausted.

# 2 Proof

# 2.1 Real

Let $c = \frac{m}{2^n}$ where $m$ is odd and $n > 0$ .

Induction step:

Let $z = \frac{M}{2^N}$ where $M$ is odd and $2N > n > 0$ .

Then $z^2 + c = \frac{M^2 + 2^{2N - n} m}{2^{2N}}$ in lowest terms because $M^2$ is odd and $2^{2N - n} m$ is even, so their sum is odd.

Base case:

$c^2 + c = \frac{m^2 + 2^n m}{2^{2n}}$

It is given that $n > 0$ , then $2n \ge 2 > n \ge 1$ .

The argument works the same way with $2$ multipliers replaced by $3$ (and other prime bases) and “odd” replaced by $\not= 0\pmod{3}$ , “even” by $= 0\pmod{3}$ .

For composite bases $b$ I think (but I’m not sure) that you need $m \not = 0 \pmod{p}$ for each prime factor $p$ of $b$ .

The argument also works changing the power from $2$ to $3$ or any integer $d > 1$ .

# 2.2 Complex

To be continued…

# 3 Alternatives

Continued fractions and continued logarithms allow exact computation with rational numbers with different space/time tradeoffs, by using (implicit) rational intervals that are computed as narrowly as needed to resolve inequalities (like $|z|^2 > 2^2$ ) to a definite answer.