What possible values can the difference of squares of two Gaussian integers take?

Question

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.

Theodore Amidon · Accepted Answer

See 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, b, c, d# and the resulting values of #(a^2-b^2-c^2+d^2)# and #(ab+cd)# modulo #4# and #2# respectively: #((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 #(ab+cd)# is odd, then #(a^2-b^2-c^2+d^2) = 0, 1# or #3# modulo #4#. So the conjecture in the question is false: If the difference of squares of two Gaussian integers has an imaginary part of the form #4k+2# then the real part is of the form #4k+0#, #4k+1# or #4k+3#. Specifically not of the form #4k+2#.

Serenity Jones · Answer

undefined The difference of squares of two Gaussian integers can take any non-negative integer value. This is because when you subtract one Gaussian integer from another and then square the result, you can obtain any non-negative integer value depending on the specific Gaussian integers chosen. The difference of squares of Gaussian integers is a non-negative integer because it is always squared, resulting in a non-negative value. Therefore, any non-negative integer can be represented as the difference of squares of two Gaussian integers.