# 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+22N−nm22N in lowest terms because M2 is odd and 22N−nm is even, so their sum is odd.
Base case:
c2+c=m2+2nm22n
It is given that n>0, then 2n≥2>n≥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.