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

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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

See explanation...

Here are some possibilities:

So we can get any (real) integer as a difference of squares of Gaussian integers.

More generally:

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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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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