# [seqfan] PrimeQ and Gaussian integers

israel at math.ubc.ca israel at math.ubc.ca
Tue May 9 05:43:14 CEST 2023

```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
```