[seqfan] Re: PrimeQ and Gaussian integers

[sven-h.simon at gmx.de]

Hello,
PARI too factors in GaussianPrimes only, when the number has an imaginary
part, an easy solution is to multiply the whole number with I in this case. 
Sven 

-----Ursprüngliche Nachricht-----
Von: SeqFan <seqfan-bounces at list.seqfan.eu> Im Auftrag von
israel at math.ubc.ca
Gesendet: Dienstag, 9. Mai 2023 05:43
An: seqfan at list.seqfan.eu
Betreff: [seqfan] PrimeQ and Gaussian integers

Today I submitted a correction to A058770, but I suspect the issue might
affect other sequences as well. A058770 is "Numbers n such that n * (1+i)^n 
+ 1 is a Gaussian prime". The current Data section has 1, 2, 3, 5, 9, 
+ 19,
20, 29, 30, 68, 142, 143, 150, 159, 160, 198, 468, 782, 858, 1100, 1137,
3337, 3638, 3909, 4845, 16895, 21768, 30349, 42692, 48470, 65208, but I have
removed 160, 21768, and 65208. There is Mathematica code:

 Do[ If[ PrimeQ[ n * (1 + I)^n + 1], Print[n] ], {n, 1, 4000} ]

Although I'm not a Mathematica expert, I think the problem is this: PrimeQ,
when given an argument that is an ordinary (i.e. rational) integer, will
test whether that argument is a (rational) prime; when given a Gaussian
integer that is not an ordinary integer, it will test for a Gaussian prime. 
The trouble here is that for the three values n = 160, 21768, and 65208, n
* (1+i)^n + 1 is a (rational) prime == 1 (mod 4), and therefore by Fermat is
not a Gaussian prime. The point is that PrimeQ does not see the I in its
input, only the result of expanding out the (1 + I)^n, which is a rational
integer when n is even, and so it only tests for a rational prime. I think
the fix would be to insert the option GaussianIntegers -> true in the call
to PrimeQ.

Cheers,
Robert

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