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Wednesday, March 28, 2018

More playing with square root of three

Number theory is focused on integers. And mainly am focused on number theory. But there is math that I find myself just doing for the fun of it, which goes into approximation. But really only because is just another demonstration of some of my math discovery. Also though is kind of a reminder of why x2 - Dy2 = 1 had historical importance. As solutions could be used to approximate square roots.

With my research noticed special cases with my BQD Iterator for two and three, where could use to get easy approximations. And is just kind of fun for me and even relaxing to advance. Last post was this one where ended up with sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

And my first post was this one for my reference.

So need the BQD Iterator, with D = -3, to advance some more:

u2 - 3v2 = 1

then it must also be true that

(u+3v)2 - 3(u+v)2 = -2

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(362x + 627y + 3(209x + 362y))2 - 3(362x + 627y + 209x + 362y)2= -2

Which is: (989x + 1713y)2 - 3(571x + 989y)2= -2

And one more iteration to be able to divide off some factors:

(989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (2702x + 4680y)2 - 3(1560x + 2702y)2= 4

So now can divide off that 4, where happens because 2 is the only prime factor of D+1, where is every two iterations, which is why is a special case. Wonder if will always get a cycle if D+1 only has 2 as a prime factor? Sounds like something to check with a computer. But might be mathematically provable also. Checking with computer first could test the hypothesis. Ok, so dividing off the 4.

And finally: (1351x + 2340y)2 - 3(780x + 1351y)2= 1

From which I have, sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 1351xn + 2340yn

yn+1 = 780xn + 1351yn

and x0 = 1, and y0 = 0;

So get x1 = 1351, and y1 = 780. Which is a decent approximation, and next is:

x2 = 3650401 and y2 = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Just using how much a web calculation showed. Another trick is to square, and look at what that gives. Where looking at prior post got more digits before.

And 36504012/21075602 approximately equals 3 is what my pc things are saying, so is too close for more with precision of the defaults. Have had LOTS more digits before but just playing around so not interested in fiddling with settings to find out why so few now.

Well guess at limits then of just being able to check with usual pc things. Oh well.

Well that was simple fun. Find it kind of relaxing. Just playing with numbers.


James Harris

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