Fractional Series which approach the square roots of prime numbers.

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SUMMARY

The discussion focuses on generating fractional series that approximate the square roots of prime numbers, specifically starting with the fractional seeds 2/1 and 9/4. The method involves a recursive rule where each new member is calculated as four times the current member plus the previous one, resulting in a sequence of fractions that converge to the square roots of primes. The participants have successfully identified seeds and production rules for square roots of primes from 2 to 23, and they are exploring patterns within these sequences. Additionally, they reference continued fractions and their convergents as a related concept.

PREREQUISITES
  • Understanding of continued fractions and their properties
  • Familiarity with recursive sequences and their generation
  • Basic knowledge of prime numbers and their square roots
  • Experience with mathematical sequences and series analysis
NEXT STEPS
  • Research the properties of continued fractions and their convergents
  • Explore the On-Line Encyclopedia of Integer Sequences for related sequences
  • Study the mathematical implications of recursive sequence generation
  • Investigate the relationship between prime numbers and their square roots in number theory
USEFUL FOR

Mathematicians, number theorists, and anyone interested in the properties of prime numbers and continued fractions will benefit from this discussion.

duren99109
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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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You can look up continued fractions which are similar to what you describe.
 

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