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%I #13 Nov 04 2015 00:21:09
%S 5,7,11,179,467,
%T 21195998530602981465199287343010006825031720870818843865120019360285948694390966280586508792391539752259819
%N Safe primes 2p + 1 such that p is a Fibonacci prime.
%C Same as safe primes q whose Sophie Germain prime (2q - 1)/2 is a Fibonacci number.
%C No other terms up to 2*Fibonacci(2904353) + 1, according to the list of indices of 49 Fibonacci (probable) primes in A001605.
%C In that range, the only safe Fibonacci prime is 5. Are there larger ones?
%C There are six primes 2p + 1 such that p is a Fibonacci prime, namely, a(1) through a(6). By contrast, in the same range there are only two primes 2p - 1 such that p is a Fibonacci prime, namely, 2p - 1 = 3 and 5, for p = 2 and 3. Is there some modular restriction to explain this bias in favor of 2p + 1 over 2p - 1 among Fibonacci primes p?
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_prime">Fibonacci prime</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Safe_prime">Safe prime</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>
%F a(n) = 2*A155011(n) + 1.
%e 179 is in the sequence because it is prime and (179 - 1)/2 = 89 = Fibonacci(11), which is also prime.
%t 2 * Select[Fibonacci[Range[2000]], And @@ PrimeQ[{#, 2 # + 1}] &] + 1
%Y Cf. A000045, A005384, A005385, A005478, A155011.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Nov 02 2015
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