 9:59 AM
@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.

6 hours later… 4:22 PM
Breaking News!!!

First factor of $\Large 10^{2^{14}}+1$ found!

p37 = 1702047085242613845984907230501142529 4:36 PM
Found with 'Prime95' on stage 2.
B1 = 12000000 (12e6)
B2 = 1200000000 (12e8)
sigma = 1970598063344121
Group order: 1702047085242613847472113166008277120 <37>

1 hour later… 5:43 PM
@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) 5:57 PM
@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 $21 \cdot 2 + 1$. 6:16 PM
I cannot find my paper where I printed out the smalles possible prima factor for some cyclotomic polynomials