Fibonacci primes equinumerous with the set of Natural numbers?

In summary, the conversation discusses the idea of infinite Fibonacci primes and the difficulty in proving their existence. The possibility of comparing the set of Fibonacci primes to the set of Natural Numbers and proving their infinite nature is also explored. However, it is pointed out that this task is not as easy as it seems and the difficulty lies in proving the infinite nature of subsets of prime numbers. The conversation concludes by acknowledging that the question has been answered and thanking the expert for their insights.
  • #1
fibonacci235
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1
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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  • #2
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 [itex]\aleph_0[/itex], 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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  • #3
Good points. I completely overlooked that fact. You've answered my question exactally. Thanks.
 
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1. What are Fibonacci primes equinumerous with the set of Natural numbers?

Fibonacci primes equinumerous with the set of Natural numbers are prime numbers that are also part of the Fibonacci sequence, which is a sequence of numbers where each number is the sum of the two preceding numbers (e.g. 0, 1, 1, 2, 3, 5, 8, 13, 21, etc.). These primes are equinumerous with the set of Natural numbers, meaning that there is a one-to-one correspondence between the two sets.

2. How are Fibonacci primes equinumerous with the set of Natural numbers determined?

Fibonacci primes equinumerous with the set of Natural numbers are determined by finding prime numbers that are also part of the Fibonacci sequence. This can be done by starting with the first two numbers in the sequence (0 and 1) and continuing the sequence by adding the two previous numbers until a prime number is reached.

3. Are there an infinite number of Fibonacci primes equinumerous with the set of Natural numbers?

It is currently unknown if there is an infinite number of Fibonacci primes equinumerous with the set of Natural numbers. However, based on the infinite nature of the Fibonacci sequence and the infinite supply of prime numbers, it is likely that there are an infinite number of these primes.

4. What is the largest known Fibonacci prime equinumerous with the set of Natural numbers?

The largest known Fibonacci prime equinumerous with the set of Natural numbers is 267904766007362. This number is part of the Fibonacci sequence and has been proven to be a prime number. However, it is possible that there are larger Fibonacci primes that have not been discovered yet.

5. What is the significance of Fibonacci primes equinumerous with the set of Natural numbers?

Fibonacci primes equinumerous with the set of Natural numbers have significance in both mathematics and computer science. In mathematics, they provide insight into the distribution of prime numbers and the behavior of the Fibonacci sequence. In computer science, they are used in various algorithms and applications, such as cryptography and data compression.

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