Primes

1 to 100

$$S = \{2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 37, 113, 28657, 514229, \\ 557, 2417, 73, 149, 2221, 2789, 59369, 433494437, 2971215073, 953, 55945741, \\ 353, 2710260697, 4513, 555003497, 269, 116849, 1429913, 6673, 46165371073, 9375829, \\ 86020717, 157, 92180471494753, 99194853094755497, 1069, 1665088321800481, 193, 389, 3084989, 361040209\}$$

Questions

• Does this hold? $$ F{n} = F{n-1} + F{n-2} $$ $$ F{n} \nmid F_{m} \qquad \forall m < n \hspace{.25cm} \land \hspace{.25cm} n \in \mathbb{P} \hspace{.25cm} \land \hspace{.25cm} m \in \mathbb{N} $$

Yes (page 3 theorem 1)

• Is all prime number contained in the sequence?

  No!

$$ Si = \{7, 17, 11, 29, 61, 47, ...\} $$

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