# Initial values of Inkeri's primality test for Fermat numbers

Can you provide a proof or a counterexample to the claim given below ?

First , we shall give a definition of the Inkeri's primality test for Fermat numbers :

Fermat's number $$F_m=2^{2^m}+1$$ ($$m \ge 2$$) is prime if and only if $$F_m$$ divides the term $$v_{2^m-2}$$ of the series $$v_0=8$$ , $$v_i=v^2_{i-1}-2$$ .

You can run this test here .

Next , we shall formulate a claim :

Let $$a_n=62 a_{n-1}-a_{n-2}$$ with $$a_1=8$$ and $$a_2=488$$ , let $$b_n=482b_{n-1}-b_{n-2}$$ with $$b_1=22$$ and $$b_2=10582$$ , then each member of the sequences $$\{a_n\}$$ and $$\{b_n\}$$ can be used as an initial value $$v_0$$ for Inkeri's test .

Union of sequences $$\{a_n\}$$ and $$\{b_n\}$$ can be found here .

You can calculate $$a_n$$ here and $$b_n$$ here .

P.S.

Initial values for Lucas-Lehmer test can be also obtained as the union of two recurrent sequences : $$a_n=14 a_{n-1}-a_{n-2}$$ with $$a_1=4$$ and $$a_2=52$$ , and $$b_n=98b_{n-1}-b_{n-2}$$ with $$b_1=10$$ and $$b_2=970$$ . See A018844 .