Editing Gaussian integer (section)

==Unsolved problems==
[[File:gauss-primes-768x768.png|170px|thumb|The distribution of the small Gaussian primes in the complex plane]]

Most of the unsolved problems are related to distribution of Gaussian primes in the plane.

*[[Gauss's circle problem]] does not deal with the Gaussian integers per se, but instead asks for the number of [[lattice point]]s 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 {{math|1 + ''ki''}}?<ref>Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F)</ref>
*Is it possible to walk to infinity using the Gaussian primes as stepping stones and taking steps of a uniformly bounded length? This is known as the [[Gaussian moat]] problem; it was posed in 1962 by [[Basil Gordon]] and remains unsolved.<ref>{{cite journal |last1= Gethner |first1= Ellen |last2= Wagon |first2= Stan |author2-link= Stan Wagon |last3= Wick |first3= Brian |doi= 10.2307/2589708 |issue= 4 |journal= [[The American Mathematical Monthly]] |mr= 1614871 |zbl=0946.11002 |pages= 327–337 |title= A stroll through the Gaussian primes |volume= 105 |year= 1998|jstor= 2589708 }}</ref><ref>{{cite book |last=Guy |first=Richard K. |author-link=Richard K. Guy |title=Unsolved problems in number theory |publisher=[[Springer-Verlag]] |edition=3rd |year=2004 |isbn=978-0-387-20860-2 |zbl=1058.11001 |pages=55–57}}</ref>