User:Reddwarf2956/sandbox

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Geocentric radius
The distance from the Earth's center to a point on the spheroid surface at geodetic latitude $$(latitude)\,\!$$ is:


 * $$R=R((latitude))=\sqrt{\frac{(a^2\cos(latitude))^2+(b^2\sin(latitude))^2}{(a\cos(latitude))^2+(b\sin(latitude))^2}}\,\!$$

where $$a = 6378.1370$$ (equatorial radius in kilometers) and $$b = 6356.7523$$ (polar radius in kilometers).

If (latitude) = tau / 4 - 1 then 6371.93128 kilometers.

$$_0\tau = 6.283185307179$$
User:Reddwarf2956/Ratio_of_circumference_to_radius A fundamental mathematical constant which has it [decimal expansion] approximately equal to 6.283185307179586476925286766559005768394338798750211641949889184615632... the ratio of a circle's [circumference] to its [radius] has no formal symbolism. However, several individuals have recommended that it be symbolized by Greek letter τ (/tau/) and other characters of lower popularity.

The value of τ = 2π and is as seen in A019692 of the OEIS and the continued fraction is found in A058291 τ = 6 + 1/(3 + 1/(1 + 1/(1 + 1/(7 + ...)))).

It has been observed that the value of τ is more mathematical fundamental than the constant [π]. Because the functional use of the [radius] and [radians], the dependency of the diameter to 2 times the radius makes π := τ/2, and the number of radians in a complete circle τ. A measure of an angle using a factional [turn] in units of τ is simple to use and understand because of its [one to one correspondence] with the number of radians. Most Scientific applications in mathematics and physics which use the constant [π] or [/pi/] can be converted to measurements in τ using the equation π = τ/2.

The constant tau is used in programming languages as names like twopi, two_pi, TWOPI, TWO_PI, ... for example FORTRAN: http://naif.jpl.nasa.gov/pub/naif/toolkit_docs/FORTRAN/spicelib/twopi.html. Or, as here a way numbering software versions: https://en.wikipedia.org/wiki/Pugs#Version_numbering.

PariGP program: log(round(exp(Tau*sqrt(652))))/sqrt(652)-Tau

Version numbering
The major/minor version numbers of Pugs converges to 2π (being reminiscent of TeX and METAFONT, which use a similar scheme); each significant digit in the minor version represents a successfully completed milestone. The third digit is incremented for each release. The current milestones are:
 * 6.0: Initial release.
 * 6.2: Basic IO and control flow elements; mutable variables; assignment.
 * 6.28: Classes and traits.
 * 6.283: Rules and Grammars.
 * 6.2831: Type system and linking.
 * 6.28318: Macros.
 * 6.283185: Port Pugs to Perl 6, if needed.

It is at the OEIS as A058291, and A019692. A simple search on Google will find many, many more examples of the constant tau by other names and used as 6.2831853.... And, tau is mathematically consistent. Yet here on Wikipedia, there is a pious bias against tau or anything that looks like tau in value. Every where I look for the constant there is no references back to tau, two pi, or even 6.2831853. They all redirect back to pi or do not exist. The above commentators reference tau and know what it is and what its value is, yet you expect me to talk about it as if it was numerology, sorry if any number here is consider numerology that is pi because as a Wikipedia topic it has the lack of consistency. And, this Wikipedia topic "pi" does not reference any of these references above or any of many other places that I did not state. In fact there is the act of hijacking 2pi and saying that it, pi, in a lot of places http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80. But, that does not change the problem pi has as stated here earlier, it is inconsistent in that it does not use radius for consistency sake. And, no one wants to agree that tau is not the issue here, it is the unnecessary bias against tau because it is mathematically a better constant to deal with things like pi. Just look at how both pi and tau is used in spherical coordinates. On the x and y axes the use of the maximum value is tau. The maximum angle from the positive z axis is pi. Yet here you insist on C/d and ignore the facts. Just do a search for the number of times an even and odd multiples of pi are used in the period of functions, or just in functions here http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80. Now pi is not the first mathematically inconsistent thing. Just look at when 1 was called prime and 2 was called not prime.

The circumference of a circle relates to one of the most fundamental and important mathematical constant in all of mathematics. This constant pi, is represented by the Greek letter $\pi$. π has the numerical value of 3.14159 26535 89793 ..., and is defined by two straight line linear correlations. The first correlation is the ratio of a circle's circumference to its diameter and equals π. While the second is the ratio of the diameter and two times the radius, and is used as to convert the diameter to radius in the same ratio as the first correlation, π. Both linear correlations combine in respect with circumference $c$, diameter $d$, and radius $r$ to become:


 * $${c}=\pi\cdot{d}={2}\cdot\pi\cdot{r}.\!$$

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. While the mathematical constant ratio of the circumference to radius, $${c}/{r} = 2\cdot\pi$$, also has many uses in mathematics, engineering, and science, it is not formally named. These uses include but are not limited to radians, computer programming, and physical constants.



The ratio of any circle's circumference to its radius is a mathematical constant equal to two times the number pi (2π). This number has a value of approximately 6.2831853 and appears in many common formulas, often because it is the period of some very common functions — sine, cosine, $τ$, and others that involve trips around the unit circle. Some individuals have proposed giving this number its own symbol and using that instead of π in mathematics notation. This proposition has been relayed in several news articles,  but has not been echoed in scientific publications nor by any scientific authority.

Advocacy
In an opinion column in The Mathematical Intelligencer, Robert Palais argued that π is "wrong" as a circle measure, and that a better value would be 2π, being the measure of the circle's circumference and the period of the sine, cosine, and complex exponential functions. He suggested a symbol like π but with three legs be used in place of 2π, demonstrating how it simplifies many mathematical formulas.

In popular culture
In 2010, Michael Hartl posted an essay called The Tau Manifesto on his personal website. In it, he proposed using the Greek letter tau ($e^{ix}$) to represent that number instead. Hartl argued that an existing symbol like $τ$ would face fewer barriers to adoption than a new symbol like the "three-legged pi" proposed in the Intelligencer. A number of news outlets reported on "Tau Day", a holiday proposed in The Tau Manifesto' for June 28 to honour the number 2π. The Royal Institution of Australia's Tau Day celebration in 2011 featured the performance of a musical work based on tau.

According to the Massachusetts Institute of Technology's Dean of Admissions Stuart Schmill, "over [the] past year or so, there has been a bit of a debate in the math universe over which is a better number to use, whether it is Pi or Tau". Thus the school chose to inform 2012 applicants whether or not they were accepted on Pi Day at what MIT called Tau Time, 6:28 pm.

Firoozbakht’s Conjecture
Firoozbakht’s Conjecture


 * 0: $$p_{n+1}^{1/(n+1)} < p_n^{1/n} \text{ for all } n \ge 1.$$


 * 1: $$p_{n+1} < p_n^{1+\frac{1}{n}}\text{ for all } n \ge 1,$$


 * 2: $$\log{p_{n+1}} < \log{p_n^{1+\frac{1}{n}}}$$


 * 3: $$\log{p_{n+1}} < \left(1+\frac{1}{n}\right)\log{p_n}$$


 * 3a: $$\frac{\log{p_{n+1}}}{1+n} < \frac{\log{p_n}}{n}$$


 * 3b: $$\frac{\log{p_{n+1}}}{\log{p_n}} < \frac{1+n}{n}$$


 * 4: $$n\left(\log{p_{n+1}} - \log{p_n}\right) < \log{p_n}$$


 * 5: $$\log{p_{n+1}} - \log{p_n} < \frac{\log{p_n}}{n}$$


 * 5a: $$\frac{p_{n+1}}{p_n} < p_n^{1/n}$$


 * 5b: $$\frac{p_{n+1} - p_n}{p_n}< p_n^{1/n} - 1 = P(n)$$ with
 * $$P(n) = \left(p_n^{1/n} - 1\right) = \left(\frac{p_np_n^{1/n} - p_n}{p_n}\right) = \left(\frac{p_n^{1+1/n} - p_n}{p_n}\right)$$


 * 5c: $$g_n = p_{n+1} - p_n< p_n\left(p_n^{1/n} - 1\right)$$


 * 5d: $$g_n = p_{n+1} - p_n< p_nP(n)$$


 * 6: $$\frac{\log{p_{n+1}} - \log{p_n}}{\log{p_n}} < \frac{1}{n}$$


 * 6a: $$n < \frac{\log{p_n}}{\log{p_{n+1}} - \log{p_n}}$$


 * 7: $$\frac{\log{p_{n+1}}}{\log{p_n}} -1 < \frac{1}{n}$$


 * 8: $$\frac{\log{p_{n+1}}}{\log{p_n}} -\frac{1}{n} < 1$$


 * 9: $$n\log{p_{n+1}} - \log{p_n} < n\log{p_n}$$


 * 9a: $$\frac{n\log{p_{n+1}} - \log{p_n}}{\log{p_n}} < n$$


 * 9b: $$\frac1{n} < \frac{\log{p_n}}{n\log{p_{n+1}} - \log{p_n}}$$


 * 9c: $$\frac{n\log{p_{n+1}} - \log{p_n}}{n} < \log{p_n}$$


 * 10: $$\pi(x * x^\theta) - \pi(x) \ge 1 \text{ for }x \ge 2\text{ with }\theta = \frac{1}{\pi(x)}$$


 * 11: $$\text{li}(x * x^\theta) - \text{li}(x) \ge 1 \text{ for }x \ge 2\text{ with }\theta = \frac{1}{\text{li}(x)}$$


 * $$\frac{x}{\log{x}-1} \le \pi(x) \le \frac{x}{\log{x}-1.1}$$ with $$x \ge 60184$$


 * $$\frac{\log{x}-1.1}{x} \le \frac{1}{\pi(x)} \le \frac{\log{x}-1}{x}$$


 * $$\log{x}\frac{\log{x}-1.1}{x} \le \frac{\log{x}}{\pi(x)} \le \log{x}\frac{\log{x}-1}{x}$$


 * 12:$$\log{p_{n+1}} - \log{p_n} < \log{p_n}\frac{\log{p_n}-1.1}{p_n} \le \frac{\log{p_n}}{\pi(p_n)} \le \log{p_n}\frac{\log{p_n}-1}{p_n}$$


 * 13:$$\log{p_{n+1}} < \log{p_n}\frac{\log{p_n}-1.1}{p_n} + \log{p_n} \le \frac{\log{p_n}}{\pi(p_n)} + \log{p_n} \le \log{p_n}\frac{\log{p_n}-1}{p_n} + \log{p_n}$$


 * 13a:$$\exp{\left(\log{p_{n+1}}\right)} < \exp{\left(\log{p_n}\frac{\log{p_n}-1.1}{p_n} + \log{p_n}\right)} \le \exp{\left(\frac{\log{p_n}}{\pi(p_n)} + \log{p_n}\right)} \le \exp{\left(\log{p_n}\frac{\log{p_n}-1}{p_n} + \log{p_n}\right)}$$


 * 13b:$$p_{n+1} < p_n^{1-\frac{1.1}{p_n}}e^{\frac{(\log{p_n})^2}{p_n}} \le p_np_n^{1/n} \le p_n^{1-\frac{1}{p_n}}e^{\frac{(\log{p_n})^2}{p_n}}$$


 * 13c:$$p_{n+1} < p_n^{1-\frac{1.1}{p_n}}e^{\frac{(\log{p_n})^2}{p_n}} \le p_n^{1+1/n} \le p_n^{1-\frac{1}{p_n}}e^{\frac{(\log{p_n})^2}{p_n}}$$


 * 14:$$\log{p_{n+1}} < \log{p_n}\left(\frac{\log{p_n}-1.1}{p_n} + 1\right) \le \log{p_n}\left(\frac{1}{\pi(p_n)} + 1\right) \le \log{p_n}\left(\frac{\log{p_n}-1}{p_n} + 1\right)$$


 * 15:$$\frac{\log{p_{n+1}}}{\log{p_n}} < \frac{\log{p_n}-1.1}{p_n} + 1 \le \frac{1}{\pi(p_n)} + 1 \le \frac{\log{p_n}-1}{p_n} + 1$$


 * 15a:$$\frac{\log{p_{n+1}}}{\log{p_n}} - 1 < \frac{\log{p_n}-1.1}{p_n} \le \frac{1}{\pi(p_n)} \le \frac{\log{p_n}-1}{p_n} $$

$$k (\log{p_k}-1.1) \le p_k \le k (\log{p_k}-1)$$

$$k (\log{p_k}-0.1) \le p_k + k \le k\log{p_k}$$

$$p_{k+1} \le p_k + k \le k\log{p_k}$$

$$p_{k+1} - k \le p_k \le k\log{p_k} - k$$

$$\frac{\log{(n+1)}}{\log{n}} > \frac{\log{p_{n+1}}}{\log{p_n}} > \frac{1+n}{n} > \frac{p_{n+1}}{p_n}$$

Since $$\log_b a=\frac{\log_d a}{\log_d b}$$ then $$ \log_{b^n} a = {{\log_b a} \over n} $$.

Now set $$d = e = \exp{(1)}$$, and let $$n = 1/n, a = p_{n+1}\text{, and }b = p_n$$:

Since $$\log_{p_n} p_{n+1}=\frac{\log p_{n+1}}{\log p_n}$$, then $$\log_{p_n^{1/n}} p_{n+1}^{1/{(n+1)}} = \frac{n}{n+1}\log_{p_n}p_{n+1} = \frac{n}{n+1}\frac{\log p_{n+1}}{\log p_n}$$.

Since the Maclaurin series for $$e^x$$ is,
 * $$e^x = 1 + x + \sum_{n=2}^{\infty}\frac{x^n}{n!}$$

this shows that $$e^x \ge 1 + x$$ therefore $$\log(e^x) \ge \log(1 + x)$$, so $$x \ge \log(1 + x)$$.

Two related conjectures (see Commments) are
 * $$\left(\frac{\log(p_{n+1})}{\log(p_n)}\right)^n < e$$,

which is weaker. And,
 * $$\left(\frac{p_{n+1}}{p_n}\right)^n < n*\log(n)$$ for all values with $$n > 5$$


 * $$n\left(\log{(p_{n+1})} - \log{(p_n)}\right) < \log{(n)} + \log{(\log{(n)})}$$ for all values with $$n > 5$$


 * $$\frac{\left(\log{(p_{n+1})} - \log{(p_n)}\right)}{\log{(p_n)}} < \frac{\log{(n)} + \log{(\log{(n)})}}{n\log{(p_n)}}$$ for all values with $$n > 5$$

Firoozbakht’s Conjecture 2
From:3a:
 * 1: $$\frac{\log{p_n}}{n}$$, || not sides parallel statements, || $$\frac{\log{p_{n+1}}}{1+n}$$.

$$\log{(p_n^{1/n})} = \log_{e^n}(p_n) = \frac{\log{p_n}}{n}$$, || $$\log{(p_n^{1/(n+1)})} = \log_{e^{n+1}}(p_{n+1}) = \frac{\log{p_{n+1}}}{n+1}$$.

Maximal Prime Gaps and the Ramanujan Prime Corollary
The i-th prime gap is denoted by $$g_i$$, it is the difference between the (i+1)-th and the i-th prime numbers,
 * $$g_i = p_{i + 1} - p_i.\ $$

Also we say that $$g_i$$ is a maximal gap, when $$g_j < g_i \text{ for all }j < i$$.

The Ramanujan Prime corollary has the condition of
 * $$2p_{i-n} > p_i \text{ for } i>k \text{ where } k=\pi(p_k)=\pi(R_n)\,,$$ similarly for the next Ramanujan prime,
 * $$2p_{j-(n+1)} > p_j \text{ for } j>l \text{ where } l=\pi(p_l)=\pi(R_{(n+1)})\,.$$ We note that for some value of $$i$$ we have $$i = l$$.

What we are now introducing is a combination of these ideas
 * $$G_{i,n,t} = g_i > g_j \text{ for all } i > j \text{ where } i>k=\pi(p_k)=\pi(R_n), i<l=\pi(p_l)=\pi(R_{(n+1)}) \text{ and }t$$ is the index of the t-th maximal gap.