A103222, A103223, A103224: more unclear definitions.

[Miles Englezou]

Dear all,

I'd like to bring to your attention a few more sequences with misleading definitions.

The name of A103222 is "Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm."

As in A391110, in my opinion the definitions of these sequences are misleading. 

The analog of Euler's totient function for Gaussian integers is a function phi_i : Z[i] -> Z. That is, its codomain is Z, so it doesn't make sense to talk of a 'real and imaginary part of the totient function phi(n) for Gaussian integers'. Any generalisation of Euler's totient function to an appropriate number ring O_K will always have codomain Z since it is defined as the number of elements of the unit group O_K/I for some ideal I.

A103222 ->  A103224 instead describe this:

Let Product z_i^(e_i) = n be the factorisation of n into Gaussian primes z_i. Define a function f : Z[i] -> Z[i] such that f(n) = Product (z_i - 1) z_i^(e_i - 1). Then A103222 is the real part of f(n), A103223 is the imaginary part, and A103224 is the norm.

So instead a function f has been defined which mimics the form of the Euler totient function; however, since there is already a clearly defined analog of the totient function for appropriate number fields, namely phi_K(x) = N(x) * Product_{p|x} (1 - 1/N(p)), N(x) the norm of x, it is misleading to call this "the totient function phi(n) for Gaussian integers."

I think it would be better to define f in the name, something like the definition above:

Let Product z_i^(e_i) = n be the factorisation of n into Gaussian primes z_i. Let f : Z[i] -> Z[i] be a function such that f(n) = Product (z_i - 1) z_i^(e_i - 1). a(n) is the [real part/ imaginary part/ norm] of f(n).

Kind regards,
Miles

[sven-h...@gmx.de]

Hello,

the whole topic of Eulers totient function for Gaussian integers is a bit complicated (well complex numbers…). It involves primitve roots and in the case of Gaussian numbers you can find some resources on how to calculate for example the modulo function in Gaussian integers, which you need there.

 

Can someone give an easy example of a primitive root of a Gaussian prime ? Just to make it possible to verify the results of an implementation in software. Please do not make to much effort with explanations on your side, just an example.

Thanks

Sven

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[sven-h...@gmx.de]

Hello,

sorry for the question, about primtive roots there are some OEIS sequences about the topic, and at least one  Pari program. So there is enough info already.

Thanks

Sven

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