Primality test $n \mid F_{n+1}+F_{n-1}-1$

Question

I have found information and references about the Fibonacci primality test:

$$n \mid F_{n-\left(\frac{5}{n}\right)}$$

but not about the following primality test:

$$n \mid F_{n+1}+F_{n-1}-1$$

Which I found holding true for the first $5000$ primes (see see here), while for $n \le 2000$ there is only one pseudoprime $n = 705$ (see here and here).

Originally I saw it as $n \mid F_{2n} - F_{n}$ and then replaced $F_{2n}$ to get the above.

Somebody can give me any reference and/or literature about this?

Answer 1

I now found that the wikipedia page Lucas pseudoprime has all the references where it talks about "an alternative definition of Fibonacci pseudoprime" and especially the article: Adina Di Porto; Piero Filipponi (1989). "More on the Fibonacci Pseudoprimes" (PDF). Fibonacci Quarterly. 27 (3): 232–242 (see link) has the test ((1.16) at page 234): $$n \mid L_n - 1$$ Which is the same as the above expressed with Lucas numbers.