# Is there a prime of the given form?

primes of the form $$\lfloor p\cdot \pi^n \rfloor$$ with a prime $$\ p\$$ and a positive integer $$\ n\$$ play a role. For the prime $$\ p=19543\$$ , according to my calculations, a prime of this form must satisfy $$\ n\ge 21\ 000\$$, hence have more than $$\ 10\ 000\$$ digits.

Does a prime of the form $$\lfloor 19543 \cdot \pi^n\rfloor$$ exist and if yes, what is the smallest positive integer $$\ n\$$ doing the job ?

Simple brute force search in Maple reveals that smallest positive integer $$n$$ is $$25536$$, the probable prime being

$$\lfloor 19543 \cdot \pi^{25536}\rfloor = \underbrace{3236982484 \dots 5580309289}_{12700 \text{ digits}}$$

Perhaps someone can verify primality (at least probable).

• Any reason to doubt Maple ? Anyway, I used the Miller-Rabin-test with $10$ random bases and the number passed it, hence is probably prime. Mar 24, 2019 at 12:31
• Do you know whether it uses the Miller-Rabin-test or the BPSW-test ? Mar 24, 2019 at 12:35
• The latter in fact is extremely reliable. Although infinite many counterexmaples are expected, noone could construct or find one.To prove the number to be prime however will not be easy (Primo is probably the best way). Mar 24, 2019 at 12:36
• Good question, the documentation mentions "one strong pseudo-primality test and one Lucas test", and Miller-Rabin wiki mentions Maple uses it, but I'm not sure how that can be trusted. I'm not familiar with BPSW-test but perhaps it could be the "one strong" test.
– Sil
Mar 24, 2019 at 12:41
• Then it probably IS the BPSW test, it combines two tests as you mention. Mar 24, 2019 at 12:42