Fibonacci primes equinumerous with the set of Natural numbers?

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SUMMARY

The discussion centers on the belief that Fibonacci primes are infinite, yet currently lacks a proof for their infinitude. Participants explore the possibility of comparing Fibonacci primes to the set of Natural Numbers to demonstrate both have cardinality aleph null. The conversation also touches on other prime subsets, such as Mersenne, Sophie Germain, and Wilson primes, questioning the implications of their cardinality. The difficulty in proving the infinitude of these sets is highlighted, with references to Euclid's established proof for the infinitude of prime numbers.

PREREQUISITES
  • Understanding of cardinality, specifically aleph null
  • Familiarity with prime number classifications, including Fibonacci, Mersenne, and Sophie Germain primes
  • Basic knowledge of mathematical proofs and their significance in number theory
  • Awareness of Euclid's proof regarding the infinitude of prime numbers
NEXT STEPS
  • Research the properties and definitions of Fibonacci primes
  • Study the concept of cardinality in set theory, focusing on aleph null
  • Examine existing proofs for the infinitude of various prime subsets
  • Explore advanced number theory techniques for proving the infinitude of specific prime sets
USEFUL FOR

Mathematicians, number theorists, and students interested in the properties of prime numbers and the complexities of mathematical proofs regarding their infinitude.

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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