Unsolved Problems
Most of the unsolved problems are related to the repartition in the plane of the Gaussian primes.
Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.
There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:
The real and imaginary axes have the infinite set of Gaussian primes 3, 7, 11, 19, ... and their associates. Are there any other lines that have infinitely many Gaussian primes on them? In particular, are there infinitely many Gaussian primes of the form 1+ki?
Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of bounded length? This is known as the Gaussian moat problem; it was posed in 1962 by Basil Gordon and remains unsolved.
Read more about this topic: Gaussian Integer
Famous quotes related to unsolved problems:
“Play permits the child to resolve in symbolic form unsolved problems of the past and to cope directly or symbolically with present concerns. It is also his most significant tool for preparing himself for the future and its tasks.”
—Bruno Bettelheim (20th century)
“The child knows only that he engages in play because it is enjoyable. He isn’t aware of his need to play—a need which has its source in the pressure of unsolved problems. Nor does he know that his pleasure in playing comes from a deep sense of well-being that is the direct result of feeling in control of things, in contrast to the rest of his life, which is managed by his parents or other adults.”
—Bruno Bettelheim (20th century)