@Peter hello

@Mathphile You see, I have plenty of new projects.

that's great

I have been lately busy with a lot of my computer science projects and classes at university

Currently, I try to improve the solution $$466 369 719 316$$ which is the first of $13$ consecutive non-squarefree numbers. I wonder whether this is the smallest solution.

10^2^14+1

Found with 'Prime95' on stage 2.

Group order: 1702047085242613847472113166008277120 <37>

@MartinHopf You have written history ! I currently search for a large prime : $\phi_{5207}(10^6+n)$. Chances should be good because small factors are impossible, nevertheless, I Passed $n=1800$ without success.

Only PRP of this cyclotomic polynomial I know is Phi(5207,1365)

@Peter Today I spend free coffe and cakes for all around!

I only found some very small prime factors of generalized fermat numbers and a solution for a problem a user proposed (but probably not the smallest). I had neither luck with finding a large prime nor with finding significant factors.

I could partially answer why $\phi_n(n-21)$ is a good prime generator. The prime factors upto $11$ occur very very rare. Try to work it out yourself.

here is the question

@Peter Interesting, Heegner numbers seems to appear $21 \cdot 2 + 1$.

I cannot find my paper where I printed out the smalles possible prima factor for some cyclotomic polynomials