Ed Pegg Jr., March 15, 2004

I learned about the complex plane
at about the same time I learned of the quadratic
formula.
With that, I knew x^2 - 2 x + 17 had a solution in complex
numbers.
I didn't come across complex numbers again until I saw the Mandelbrot set
for the first time.

Figure 1. The Mandelbrot Set, colored by average differences
between iterations.

After that, I was fascinated by the complex plane. I remember
a fellow enthusiast
showing me the basic Mandelbrot set. "I managed to get it to
render in
just 8 hours!" Computers have sped up tremendously since
then. There are dozens of free programs for Mandelbrot set
explorations, allowing real-time zooms into all manner of beautiful
structures. I can particularly recommend Fractal Extreme
and Fractint.

Outside of fractals, one rarely sees complex numbers in the
newspapers. For example, the number 2^{20996011} - 1
recently made headlines as the new largest prime number, as part of the
Great Internet Mersenne
Prime Search. A lesser headline was made by the number 5359•2^{5054502}+1,
found by the distributed
attack on the Sierpinski Problem. The Collatz problem
has been tested out to 262•2^{50} by the 3x+1
search. Many other distributed problems are listed by Aspenleaf
Concepts, Inc.

Similar searches for Gaussian Integers are unknown to me. For
complex numbers, the Zetagrid
project has collected the first 687 billion zeroes of the Riemann Zeta
function. 1239587702.54745 is an interesting discovery, but it
isn't a Gaussian integer. For almost every interesting number
theory question, there is a similar question in the Gaussian integers.

For example, the Fermat-Catalan
conjecture
looks for high powers summing to other high powers. Ten examples
are known: 1^{p}
+ 2^{3} = 3^{2} (*p* __>__ 2), 2^{5}
+ 7^{2}
= 3^{4}, 13^{2} + 7^{3} = 2^{9}, 2^{7}
+ 17^{3} = 71^{2}, 3^{5} + 11^{3}
=
122^{2}, 33^{8} + 1549034^{2} = 15613^{3},
1414^{3} + 2213459^{2} = 65^{7},
9262^{3}
+ 15312283^{2} = 113^{7}, 17^{7} + 76271^{3}
= 21063928^{2}, and 43^{8} + 96222^{3} =
30042907^{2}. In Gaussian integers, Fred W. Helenius and
myself have found 7 examples: (8+5i)^{2} + (5+3i)^{3} =
(1+2i)^{7},
(20+9i)^{2}
+ (1+8i)^{3} = (1+i)^{15}, (5i)^3 + (3i)^7 = (34-34i)^2, (49+306i)^2 +
(1+2i)^7 = (27+37i)^3, (44+83i)^2
+ (31+39i)^3 = (5+2i)^7, (19+36i)^2
+ (1-i)^13 = (9+8i)^3, and
(2+i)^4
+ (1+i)^9 = (5+4i)^2.

The ABC
conjecture looks for sums a+b=c where a, b, and c are all divisible
by high powers. For example 7^{3} + 3^{10} = 2^{11}•29.
This problem could be readily adapted to the Gaussian integers.

Beal's
Conjecture
is false for Gaussian integers. Fred W. Helenius found a single
counterexample: (-2+i)^3 + (-2-i)^3 = (1+i)^4. Whether there are
more counterexamples, I don't know.

A Sierpinski
Number is a number k such that k•2^{n}
+ 1 is always
composite. I discovered that 10+3i is a Sierpinski number.
On the other hand, for 26+i, you won't find a prime until (26+i)•2^{4890}
+ 1. I don't know of any distributed searches for large Gaussian
primes.

The Catalan Conjecture has been solved for normal integers. Fred W. Helenius found several solutions among the Gaussians. (78+78i)^2 + (23i)^3 = i, 1 + (1-i)^5 = (1+2i)^2, i + (11+11i)^2 = (3i)^5.

Continued Fractions can represent any real number. For
example, Pi can be represented as a continued fraction. Pi = {3, 7, 15,
1, 292, 1, 1, 1, 2, 1, 3 ...}.

Pi = 3.14159265... =

Figure 2. Pi as a continued fraction.

It turns out that complex numbers can
also be represented as fractions. The best method at the moment
was devised by Asmus Schmidt. It works by careful consideration of 8
matrices.

Figure 3. The eight important matrices for complex continued fractions.

Figure 3. The eight important matrices for complex continued fractions.

A complex number, for example e^{i},
can be represented as a series of these matrices: c c v3 c c v3 v3 v3 c
c v3 v3 v3 v3 v3 c c and so on, in this case. If you multiply
these matrices together, is the result. If the top
and bottom are added
up, the fraction (1501 + 820 i)/(1501 - 820 i) can be made, which
approximately equals 0.5403023380 + 0.8414709642 i. e^{i}
starts off 0.5403023059 + 0.8414709848 i. So how does this
work? It turns out that these matrices can be used to divide up
the complex plane into 8 parts, in the first generation. After
that, each triangular region is divided into 4 parts, by adding a
circle. That part works like an Apollonian
packing. In addition, each circle is divided into 8 regions, by
adding 3 tangent circles.

Figure 4. The first generation of Asmus Schmidt's complex continued fraction method.

In the third generation, we already have
quite a few circles. This represents all the regions
representable by 3 of the matrices above.

Figure 5. The third generation of Asmus Schmidt's complex
continued fraction method.

Here is a portion of Generation 4.
We are starting to zoom in on e^{i}, positioned within the
generations as c c v3 c c, so far.

Figure 6. The fourth generation of Asmus Schmidt's complex
continued fraction method.

I wanted to look at a zoomed-in picture of
e^{i}
at generation 50 or so, but I haven't quite figured it out how to draw
all the circles.
Specifically, I need to draw three tangent circles inside a larger
circle, given 3 points of tangency. I'll find it. Just as some
things deserve more study, other things deserve a nice picture.

**References:**

ABC Conjecture page, http://www.math.unicaen.fr/%7Enitaj/abc.html.

The Beal Conjecture, http://www.math.unt.edu/%7Emauldin/beal.html.

Great Internet Mersenne Prime Search, http://www.mersenne.org/prime.htm.

Internet-based Distributed Computing Projects, http://www.aspenleaf.com/distributed/ap-math.html.

Asmus Schmidt. "Ergotic Theory of Complex Continued
Fractions", Lecture Notes in Pure
and Applied Mathematics, vol 147. p 215-276. 1993.

Seventeen or Bust, http://www.seventeenorbust.com/.

Eric W. Weisstein. "Apollonian Gasket", "Catalan Conjecture",
"Collatz Problem", "Complex Plane", "Fermat-Catalan Conjecture",
"Mandelbrot Set", "Quadratic Formula", "Sierpinski Number". From
*MathWorld*--A Wolfram
Web Resource. http://mathworld.wolfram.com/

Zetagrid, http://www.zetagrid.net/.

Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com.
He works at Wolfram Research, Inc. as the administrator of the
*Mathematica*
Information Center.