Science Forums 2

[Don Blazys]

To: Modest,

Well, Newtons method can be unreliable if the zero is too near an asymptote or extremum, and that derivative looks far too complicated and involved for such a simple function! I am now working on finding a simpler and more reliable derivative, because my calculation of the second root or "Prime generating constant" is:

2.566,543,832,171,7....

and thats using nothing but the "graph" function of a hand held calculator! As you can see, it is somewhat different from your calculation:

2.566,543,832,172,4.... ,

so it is clear that more reliable calculations are in order. There are many ways to determine derivatives, and I'm quite certain that I can find one that is more suitable for our purposes than the monstrosity that was presented to you by that online derivative finder.

Then, there is this to consider: Calculating the first logarithm to a large number of decimal places must rely on whatever "mechanism" is "built in" to the calculating machine or device. Thus, calculating the first logarithm to a large number of decimal places is probably the "hard part", and may require some "trial and error" work. However, calculating the second logarithm to a very high degree of accuracy can indeed be accomplished using the "Taylor series expansion of natural logarithms" because taking the first logarithm puts the value of the number in a range between +1 and -1.

By the way, I found some of my old notes on how I calculated the constant:

2.566,543,832,171,388,844,467,529...

to 24 decimal places, but I need not post it now because it is essentially the same method that CraigD used to calculate my prime generating constant to 500 decimal places.

Anyway, thanks for all the hard work that you did so far. You know, if I'm right, then it's an incredible discovery, and if I'm wrong, then it's an incredible "mathematical curiosity". Either way, the word "incredible" applies!

Don.