Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
General Math
Differences between Gaussian integers with norm 25
Reply to thread
Message
[QUOTE="andrewkirk, post: 6011883, member: 265790"] The Gaussian integers you list have norm 5, not 25. The norm of a complex number is the [I]square root [/I]of the sum of squares of the real and complex components. More [URL='https://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm_of_a_complex_number']here[/URL]. For any positive integer n there will be at least four Gaussian integers with that norm: ##n\,i,-n\,i,n,-n##. There will be additional Gaussian integers with that norm iff n is the largest number in a [URL='https://en.wikipedia.org/wiki/Pythagorean_triple']Pythagorean Triple[/URL]. Where that is the case, say ##n^2 = m^2 + k^2##, there are eight additional Gaussian integers with norm n, being ##\pm m\pm k\,i## and ##\pm k\pm m\,i##. So studying Gaussian integers with the property you seek is equivalent to studying Pythagorean Triples, on which there has been plenty of work. The linked wiki article on Pythagorean Triples has a section dedicated to their relationship to Gaussian integers [URL='https://en.wikipedia.org/wiki/Pythagorean_triple#Relation_to_Gaussian_integers']here[/URL]. The article proves that there are infinitely many Pythagorean Triples, and hence infinitely many integers that have at least twelve Gaussian integers with that norm. I think, based on the following statement in the wiki under heading 'General Properties', that it is possible for an integer to be the largest number in more than one Pythagorean triple, in which case there would be at least 4 + 2 x 8 = 20 Gaussian integers with that norm: [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
General Math
Differences between Gaussian integers with norm 25
Back
Top