The first few Fibonacci primes are 2, 3, 5, 13, 89.
Fibonacci primes with thousands of digits have been found but it is not known whether there are infinitely many (Wikipedia, see link).
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No. For example: 4181 / 37 = 113 so it can't be prime. But 4181 is the first composite number in the Fibonacci sequence with a prime index. ;)
Fibonacci's accomplishments were the creation of the Fibonacci number sequence and he also was responsible for the spreading of the Hindu-Arabic numbering system.
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No, there is no largest prime number. Also 57 is not a prime.No, there is no largest prime number. Also 57 is not a prime.No, there is no largest prime number. Also 57 is not a prime.No, there is no largest prime number. Also 57 is not a prime.
Yes - 5 is the fifth Fibonacci number.