LBg

See http://oeis.org/A001175: A001175 Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n. 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, 16, 30, 48, 24, 100, 84, 72, 48, 14, 120, 30, 48, 40, 36, 80, 24, 76, 18, 56, 60, 40, 48, 88, 30, 120, 48, 32, 24, 112, 300, 72, 84, 108, 72, 20, 48, 72, 42, 58, 120, 60, 30, 48, 96, 140, 120, 136 for n=1,2,... corresponding to your base=2,3,... There is also a table of values for n = 1..10000. See also: http://math.ca/crux/v23/n4/page224-241.pdf So no programming is needed in this case :) . Unless, of course, you want values beyond infinity...

Don, you flatter me. I am not a mathematician, neither great nor small, but I enjoy creating efficient programming solutions to mathematical problems. /LBg

I have estimated the asymptotic behaviour of w(10^n) as n goes to infinity and found it to be 0.640362740055367 * 10^n. See attachment for details. /LBg wAsymptotic.pdf

Don, I have no website of my own and have no need create one either, sorry! /LBg

"Mommy, how many regular figurative numbers are there? I feel so utterly disregarded, poor and helpless!" "Here, my little girl, Mommy will explain to you in a simple way that even you as a small child will understand:" [math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-\pi^{2}\right)}\right) [/math] where: [math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(\left(\pi^e+e^{\frac{-\pi}{2}}+4+\frac{5}{16}\right)*\left(\ln\left(x\right)\right)^{-1}+1\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math] "Now you see, don't you?"

Well, of course not. In science you formulate your hypothesis and test it against available data. If it fits, then it is accepted. Later new data, and new understanding comes along, which leads to a new hypothesis in accordance with the new data. This does not mean that the previous hypothesis was "wrong", it was the best that could be done at the time. Isaac Newton, a genius of his time, formulated his theory of gravity, later superseeded by Einsteins relativity version. And it is quite possible that Einsteins theory will replaced by something else in the future. That's how science progresses, without labelling people as being "wrong".

137 is a prime number 137 is a prime of the form 8n+1 137 is a prime of the form 3n-1 137 is a prime of the form 6n-1 137 is a prime of the form 2n+3 137 is a prime of the form 30n-13 137 is a prime of the form x^2+101y^2 (x=6, y=1) The sum of digits of 137 is a prime (namely 11) 137 is the lesser of a pair of prime twins 137 is a prime p such that 3p-2 is prime 137 is a prime p such that 2p+1 is composite 137 is a number n such that (10+n!)/10 is prime 137 is a number n such that 6n-1, 6n+1 are twin primes 137 is a number n such that (13^n - 1)/12 is prime. 137 remains prime if any digit is deleted 137 is not the sum of 2 primes 137 is the number of primes between 2^10 and 2^11 Fib(137) is a prime number 137 is odd but not divisible by 5 137 is the sum of 4 positive cubes in one or more ways 137 is both the sum of two nonzero squares and the difference of two nonzero squares 137 = 4^2 + 11^2 and 4/137=0.0291970802919708..., 11/137=0.080291970802919708... same digits! 137 occurs in the pythagorean triples (105, 88, 137) and (137, 9384, 9385) and no other 137 is a number of the form x^2 + xy + 2y^2, (x=1, y=8, and 1+8 = 9 = 3^2) 137 is a number of the form x^2 + 2*y^2, (x=3, y=8) 137/60 = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 137 is the number of restricted hexagonal polyominoes with 5 cells 137 is a number n such that Mordell's equation y^2 = x^3 + n has no integral solutions

Not by adding two squares, but by subtracting: 34^2 - 11^2, 42^2 - 27^2, 62^2 - 53^2, 106^2 - 101^2, 174^2 - 171^2, 518^2 - 517^2

Here are a few more that have equally interesting properties (some of them using fewer than 4 numerals): 1/101 = 0.00990099009900990099...; 100/101 = 0.9900990099009900... 1/110 = 0.00909090909090909090...; 109/110 = 0.9909090909090909... 1/111 = 0.00900900900900900900...; 110/111 = 0.9909909909909900... 1/273 = 0.00366300366300366300...; 272/273 = 0.99633699633699699633... 1/303 = 0.00330033003300330033...; 302/303 = 0.99669966996699669966... 1/1001= 0.00099900099900099900...; 1000/1001 = 0.999000999000999000999... Note: 273 is the boiling point of water in degrees Kelvin! Note: 273/2 = 136.5, almost 1/FSC!

I know you don't, that is why I remarked that 0.61803399*1000 rounded to an integer is by the rule of rounding 618, not 620 as you claim. Well, actually, you do not claim it, you say "relating it to" but that is neither an exact nor a scientific statement.

Why not use an approximation which is both more accurate and has lower denominator, such as 61/57 (both are primes!), 76/71, 91/85, 107/100 (10^2!), and many more ? Why is that interesting? The "lower Golden Ration" is 0.61803399, 1000x this is 618, not 620.

The initial hypothesis was that the counting function could be used to give a value of [math]\alpha^{-1}=137.035999084(51)[/math] which would become accurate as more data from the counting was obtained. When it turns out that more counting data does not support this hypothesis it is abandoned and another one is constructed, using the concept of a "running constant". I don't see why [math]\alpha^{-1}=137.035999084(51)[/math] which is a number with a stated uncertainty should be regarded as a true constant whereas [math]\alpha^{-1}\approx 137.03605(5)[/math] which is another number with a stated uncertainty should be called "running". In any case, a given formula connecting the counting data with [math]\alpha^{-1}[/math] can only give a single value for [math]\alpha^{-1}[/math], not a range of values. It is not a simplified version, it is more complicated in that [math]x[/math] is now included in the expression for [math]\alpha[/math]. Yes, you are right, the new function matches the current counting data better. /LBg

Yes, I am the fellow that helped Don calculate w(10^15). I have extended the calculations in Don's paper, and made some investigations on my own. The results indicate that the presence of the "fine structure constant" in these data is rather speculative. It is even doubtful that the form of the equation is the best one for approximating the data. I have attached my investigations in PDF form. /Lars Investigation.pdf