What possible values can the difference of squares of two Gaussian integers take?
The difference of squares of any two integers can take the form 4n+k for any integer n and k in {0, 1, 3} . Specifically, the difference of squares of two integers cannot be of the form 4n+2 .
Is there a simple characterisation of possible differences of squares for Gaussian integers, i.e. complex numbers of the form m+ni , where m, n are integers ?
Conjecture: Any Gaussian integer of the form m+2ni where m, n are integers is expressible as the difference of two squares of Gaussian integers.
The difference of squares of any two integers can take the form
Is there a simple characterisation of possible differences of squares for Gaussian integers, i.e. complex numbers of the form
Conjecture: Any Gaussian integer of the form
1 Answer
See explanation...
Explanation:
Here are some possibilities:
-
(n+1)^2-n^2 = 2n+1 -
(n+1)^2-(n-1)^2 = 4n -
((n+1)+ni)^2 - (n+(n+1)i)^2 = 4n+2
So we can get any (real) integer as a difference of squares of Gaussian integers.
More generally:
(a+bi)^2-(c+di)^2 = (a^2-b^2-c^2+d^2)+2(ab+cd)i
Consider the various possible combinations of odd and even
((a_2, b_2, c_2, d_2, (a^2-b^2-c^2+d^2)_4, (ab+cd)_2),(0,0,0,0,0,0),(0,0,0,1,1,0),(0,0,1,0,3,0),(0,0,1,1,0,1),(0,1,0,0,3,0),(0,1,0,1,0,0),(0,1,1,0,2,0),(0,1,1,1,3,1),(1,0,0,0,1,0),(1,0,0,1,2,0),(1,0,1,0,0,0),(1,0,1,1,1,1),(1,1,0,0,0,1),(1,1,0,1,1,1),(1,1,1,0,3,1),(1,1,1,1,0,0))
So if
So the conjecture in the question is false: If the difference of squares of two Gaussian integers has an imaginary part of the form