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If the Fibonacci sequence is denoted by F(n), where n is the first term in the sequence then the following equation obtains for n = 0.
If you start with 1, the common factors are 1 and 3. If you start with zero, as Fibonacci did, the common factor is 1.
Assume the question refers to the standard Fibonacci sequence where the first two numbers are 0 and 1.The sequence progresses :- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 - which is the number required.
Leonardo Fibonacci first recorded his sequence in his book Liber Abaci, which was published in 1202.
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. ;)