Fibonacci primes equinumerous with the set of Natural numbers?

AI Thread Summary
The discussion centers on the belief that Fibonacci primes are infinite, though no proof currently exists to support this. Participants question whether comparing the set of Fibonacci primes to the set of Natural Numbers could demonstrate both have cardinality aleph null, implying their infinitude. They also explore the difficulty in proving the infinitude of various prime subsets, noting that while Euclid proved the infinitude of primes, this does not automatically extend to all subsets. The conversation highlights the complexity of establishing the infinitude of specific prime categories, with examples illustrating that not all subsets of primes are infinite. Ultimately, the challenge remains in proving the infinitude of Fibonacci and other specific prime sets.
fibonacci235
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I believe that Fibonacci primes are infinite. Currently there is no proof that there is an infinite number of Fibonacci primes. I was wondering why we couldn't compare the set of Fibonacci primes to the set of Natural Numbers and demonstrate that both have cardinality aleph null? Indeed, why couldn't we do that for Mersenne primes, Sophie Germain primes, Wilson primes etc.

Wouldn't that imply that these sets of prime numbers are infinite? I'm assuming if it were that easy someone who have demonstrated that by now. Clearly, that is not the case, so I am wondering why it is so difficult to prove that these sets of Prime numbers are infinite. After all, Euclid demonstrated that the set of primes is infinite; wouldn't that imply that its subsets would be infinite too?
 
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fibonacci235 said:
I believe that Fibonacci primes are infinite. Currently there is no proof that there is an infinite number of Fibonacci primes. I was wondering why we couldn't compare the set of Fibonacci primes to the set of Natural Numbers and demonstrate that both have cardinality aleph null? Indeed, why couldn't we do that for Mersenne primes, Sophie Germain primes, Wilson primes etc.

If we show that both have cardinality \aleph_0, then we would indeed have shown that there are an infinite number of them. But it's as difficult to prove.

Wouldn't that imply that these sets of prime numbers are infinite? I'm assuming if it were that easy someone who have demonstrated that by now. Clearly, that is not the case, so I am wondering why it is so difficult to prove that these sets of Prime numbers are infinite. After all, Euclid demonstrated that the set of primes is infinite; wouldn't that imply that its subsets would be infinite too?

Not at all. For example {57} is a subset of the primes, but it isn't infinite. Or the "even prime numbers" are also a subset, but this set is finite.
 
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Good points. I completely overlooked that fact. You've answered my question exactally. Thanks.
 
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