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# 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=m2n where m is odd and n>0.

Induction step:

Let z=M2N where M is odd and 2N>n>0.

Then z2+c=M2+22Nnm22N in lowest terms because M2 is odd and 22Nnm is even, so their sum is odd.

Base case:

c2+c=m2+2nm22n

It is given that n>0, then 2n2>n1.

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.