Fractional Series which approach the square roots of prime numbers.

In summary, the conversation discusses the generation of new members in a sequence by multiplying the previous number by 4 and adding the previous number. This process generates the progression of fractions that represent the irrational root of sqrt(5). The conversation also mentions finding the seeds and rules for the square roots of primes 2 through 23 and looking for patterns. The conversation suggests looking into continued fractions and provides links to further resources on the topic.
  • #1
duren99109
1
0
Such as sqrt 5: (2.236067977...)
Start with the fractional seeds 2/1, 9/4,...

New members are generated (both numerators and denominators) by the rule new member = 4 times the current plus the previous.

Which generates the progrssion 2/1, 9/4, 38/17, 161/72, 682/305, 2889/1292, 12238/5473, 51841/23184... (and this process keeps generating the digits of the irrational root)

Have found the seeds and rules of production for the sqrts of primes 2 through 23 and trying to find patterns...

anybody either (1) know about this stuff, or (2) find it of interest?

I would appreciate any thoughts on the matter...
 
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  • #2
You can look up continued fractions which are similar to what you describe.
 
  • #3

1. What are fractional series that approach the square roots of prime numbers?

A fractional series is a sequence of numbers created by dividing two consecutive terms. In the case of approaching the square roots of prime numbers, the series would be created by dividing a prime number by its consecutive number.

2. What is the significance of approaching the square roots of prime numbers using fractional series?

The significance of this approach is that it can be used to approximate the square roots of prime numbers, which are typically irrational numbers. This can be useful in various mathematical and scientific calculations.

3. How accurate is this method in approximating the square roots of prime numbers?

The accuracy of this method depends on the length of the fractional series used. The longer the series, the more accurate the approximation will be.

4. Can this method be applied to all prime numbers?

Yes, this method can be applied to all prime numbers. However, the accuracy of the approximation may vary depending on the size of the prime number.

5. Are there any limitations to using fractional series to approach the square roots of prime numbers?

One limitation is that the accuracy of the approximation decreases as the size of the prime number increases. Additionally, this method can only approximate the square root of a prime number, it cannot give an exact value.

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