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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 .

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