Another Ramanujan Trick

[Gerson W. Barbosa]

(10-03-2019 09:44 PM)Valentin Albillo Wrote:  

(10-03-2019 07:58 PM)Gerson W. Barbosa Wrote:  I thought the associated polynomial is interesting just the same. It arises from noticing that the 34th power of pi is 8.00010471505E16, whose underlined digits match the first digits of pi/3.

Nice find, indeed. Did you mention it in the past ? I seem to remember having read about it in one of your old posts but I may just be mistaken.

I think I have used that in a near-integer expression, but I can't find it right now.

(10-03-2019 09:44 PM)Valentin Albillo Wrote:  

Quote:I forgot to mention that it take more than 3 minutes for the HP-71B to show the result.

To be fair, you must consider that it's not simply computing just the one root you're interested in for your approximation to Pi, but rather it's simultaneously computing all 34 complex roots to full accuracy (using 15 digits internally). By the time the 3 minutes have elapsed you have the 68 components of the 34 complex roots readily stored in complex array B(), though you then go and use just one of them.

Considering this fact, 3 minutes doesn't seem that long, it comes to less than 5.3 seconds per 12-digit root. And, perhaps, using FNROOT instead to compute just the one root you're interested in would be much faster than using PROOT to compute all 34 roots. Did you try ?

No, I didn't. But that surely would have been much faster. At first, I tried Wolfram Alpha, which does it instantaneously, but I thought the HP-71B code would fit better here. Anyway, here it is, just for the record:

Solve x^34 - (1/3)*10^12*x - 8*10^16 = 0

Best regards,

Gerson.