[seqfan] Re: Why is 265 not a strong pseudoprime to base 54?

[Neil Fernandez]

Hi Alonso,

In message <CAGyGvfUZQyYKzCQgP6wTsJ+Uuv4zZoONcJvYRKo3LE4pGGz3PQ at mail.gma
il.com>, Alonso Del Arte <alonso.delarte at gmail.com> writes

>Why is 265 not a strong pseudoprime to base 54? It's not listed in
>A020280, so there must be a good reason.
>
>54^264 = 1 mod 265, but of course that's not enough to be a strong
>pseudoprime. Since 264 = 8 * 33, I check that 54^32 = 1 mod 265. Did I
>take a wrong turn somewhere?

Yes. We need to test 54^33 rather than 54^32.

Congruences mod 265 are as follows:

54^264 = 1

54^132 = 1

54^66 = 1

54^33 = 54, so 54 is not a strong pseudoprime mod 265.

Neil