In 2008 and 2009, Denomme-Savin and Tsumura provided 2 papers providing a Primality Test for Fermat numbers based on Elliptic Curves technic:
$$ \text{Let } DST(x)= \frac{\displaystyle x^4+2x^2+1}{\displaystyle 4(x^3-x)} \ \text{ and } \ F_n = 2^{2^n}+1 $$ $$ \text{Let } x_1=F_1=\pm 5, \ x_{j+1} = DST(x_j)$$ $$ \text{If } x_{2^{n-1}} \equiv \mp 1 \pmod{F_n} \ , \ \text{then } F_n \text{ is prime.}$$
Now, can you prove the following conjecture (which makes use of a BigTree of the DiGraph):
$$\text{Let } x_1 = \pm 4 \ , \ x_{j+1} = DST(x_j) $$ $$\text{If } x_{2^{n-1}} \equiv 0 \pmod{F_n} \ , \ \text{then } F_n \text{ is prime}.$$
And, then, can you prove the following conjecture (which makes use of a Cycle of the DiGraph):
$$\text{Let } x_1 = \pm 1/3 \ , \ x_{j+1} = DST(x_j) $$ $$\text{If } x_{2^{n}-1} \equiv x_1 \pmod{F_n} \ , \ \text{then } F_n \text{ is prime}.$$
R. Denomme and G. Savin, “Elliptic curve primality tests for fermat and related primes,” Journal of Number Theory, vol. 128, no. 1, pp. 2398–2412, 2008.
Y. Tsumura, “Elliptic curve primality tests for fermat and related primes” 2009.