Gerry Myerson examined the sequence $$a_n=\gcd(2^{2^n}-1,3^{2^n}-1)$$
I am interested in the factors of $a_n$.
Since $2^{2^n}-1$ is the product of first $n-1$ Fermat numbers, and $3^{2^n}-1=2 \prod\limits_{k=0}^{n-1}{3^{2^k}+1}$, factors of $a_n$ will be in the intersection.
What is currently known to me about the factors:
For $n<20$, the factors are Fermat Primes.
For $20 \le n \le 36$, $a_n=5 \cdot 17 \cdot 257 \cdot 65537 \cdot 13631489$. $a_{39}$ is divisible by $ 2^{41} \cdot 3+1$.
Search in the intersection of known Fermat factors and known factors of generalized fermat numbers GFN'(3)=$\frac{3^{2^m}+1}{2}$ showed 19 factors of $a_n$. The format is pair $(k,n)$ with $p=k\cdot2^n+1$:
[(3, 41), (21, 41), (9, 67), (5, 127), (3, 209), (7, 320), (29, 4727), (9, 9431), (81, 12189), (81, 28285), (7, 95330), (3, 157169), (3, 213321), (3, 303093), (3, 382449), (9, 461081), (243, 495732), (3, 2145353), (3, 2478785)]
To verify the above, work modulo the factor.
Computing the actual gcd for $n=33$ will require about 12G of RAM and will take about 7 hours in a C++ ntl implementation.
Would it be possible some clever trick to reduce the memory requirement, hopefully finding unknown Fermat factor?
Another question are infinitely many primes expected to divide $a_n$?