Fractional Series which approach the square roots of prime numbers.

[duren99109]

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......

[NateTG]

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

[geraldmcgarvey]

Indeed, these fractions are the continued fraction convergents to the sqrt(5). see these "On-Line Encyclopedia of Integer Sequences" entries: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001077
and http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001076
(Numerators and Denominators of the convergents)