Does the recurrence relation
$$ a(n+1) = a(n)^2 + 1,\quad a(1)=1, $$
have a closed form solution?
I have tried hard to find it, but failed. Any ideas ?
I am particular interested in prime factors of a(n), the least n for which I did not find a prime factor, is 89. With trial division, I checked up to about 10^9 and did not find a prime factor. Is there a better method to find prime factors for such large numbers than the time-consuming trial division ?