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

OEIS: A019692

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 π. 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.

Some special angles in terms of τ

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, e^ix, 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 (τ) 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.

See also

• Turn (geometry)

External links

• Robert Dixon (Author) (18 June 2011). Pi ain't all that (flv) (YouTube).
• Vi Hart (Author) (14 March 2011). Pi Is (still) Wrong (flv) (YouTube).
• Salman Khan (11 July 2011). Tau versus Pi (flv) (YouTube). Khan Academy.
• OEIS: A019692 Decimal expansion of 2*Pi and related links at the On-Line Encyclopedia of Integer Sequences

Category:Pi

Amira, Dan (14 March 2011). "Pi Is Very Slowly and Nerdily Going Out of Style". New York. Retrieved 2012-04-03. Anthony, Sebastian (28 June 2011). "Down with pi! Today is Tau Day". ExtremeTech. Retrieved 2012-04-03. Black, Debra (28 June 2011). "Down with ugly pi, long live elegant Tau, physicist urges". Toronto Star. Retrieved 2012-04-03. Chant, Ian (28 June 2011). "Happy Tau Day!". Science Friday. Retrieved 2012-04-03. Connelly, Claire (28 June 2011). "What does Tau sound like? Pi, except better". The Advertiser (Adelaide). Retrieved 2012-04-03. Fattah, Geoffrey (4 July 2011). "University of Utah math professor starts international math movement". Deseret News. Retrieved 2012-04-03. Haught, Nancy (28 June 2011). "Tau Day today: Mathematicians show their work". The Oregonian. Retrieved 2012-04-03. Henderson, Mark (28 June 2011). "Maths mutineers say number's up for pi". The Australian. Retrieved 2012-04-03. Hudson, John (28 June 2011). "Down with Pi! The Math Nerds Behind the Tau Movement". The Atlantic Wire. Retrieved 2012-04-03. Landau, Elizabeth (28 June 2011). "In case Pi Day wasn't enough, it's now 'Tau Day' on the Internet". CNN. Retrieved 2012-04-03. Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 2011-04-03. Peddie, Clare (28 June 2011). "Moves to replace Pi with Tau". The Advertiser (Adelaide). Retrieved 2012-04-03. Shen, Anqi (29 June 2011). "Math wars: Debate sparks anti-pi day". PhysOrg. Retrieved 2012-04-03. Springmann, Alessondra (June 28, 2011). "Tau Day: An Even More Fundamental Holiday Than Pi Day". PC World. Retrieved 2011-04-03. Tagg, Cori (28 June 2011). "Happy National Tau Day, brainiacs". Deseret News. Retrieved 2012-04-03. Tovrov, Daniel (28 June 2011). "Happy Tau Day!". International Business Times. Retrieved 2012-04-03. Usigan, Ysolt (June 28, 2011). "Happy Tau Day, everybody!". Tech Talk (CBS News). Retrieved 2012-04-03. Wolchover, Natalie (June 29, 2011). "Mathematicians want to say goodbye to pi". Life's Little Mysteries (MSNBC). Retrieved 2012-04-03. "Life of pi over? 'Tau' may set calculations aright". The Times of India. June 29, 2011. Retrieved 2012-04-03. "On National Tau Day, Pi Under Attack". Fox News Channel. NewsCore. June 28, 2011. Retrieved 2011-04-03. "Tau: Is two pi better than one?". Today (BBC Radio 4). 28 June 2011. Retrieved 2012-04-03. "Your number's up: Why mathematicians are campaigning for pi to be replaced with alternate value tau". Daily Mail. June 29, 2011. Retrieved 2012-04-03.

A new mathematical constant tau (τ) has been proposed which equals two times π. Its supporters have argued that as the ratio of circle circumference to radius, this new constant is a more natural choice than π and would simplify many formulae.[1][2] While their proposals, which include celebrating 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature.[3][4] ^ Abbott, Stephen (April 2012). "My Conversion to Tauism". Math Horizons 19 (4): 34. doi:10.4169/mathhorizons.19.4.34. ^ Palais, Robert (2001). "π Is Wrong!". The Mathematical Intelligencer 23 (3): 7-8. doi:10.1007/BF03026846. ^ Hartl, Michael. "The Tau Manifesto". Retrieved 28 April 2012. ^ Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 28 April 2012.

See also

• Prime number theorem
• Andrica's conjecture, Cramér's conjecture, Firoozbakht’s Conjecture, Legendre's conjecture, and Oppermann's conjecture which have weaker but still unproven upper bounds on prime gaps.

-

Firoozbakht’s Conjecture

Firoozbakht’s Conjecture

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

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

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

$P(n)=\left(p_{n}^{1/n}-1\right)=\left({\frac {p_{n}p_{n}^{1/n}-p_{n}}{p_{n}}}\right)=\left({\frac {p_{n}^{1+1/n}-p_{n}}{p_{n}}}\right)$

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

5d: $g_{n}=p_{n+1}-p_{n}<p_{n}P(n)$

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

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

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

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

${\frac {x}{\log {x}-1}}\leq \pi (x)\leq {\frac {x}{\log {x}-1.1}}$ with $x\geq 60184$

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

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

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

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

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

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

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

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

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

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

$k(\log {p_{k}}-1.1)\leq p_{k}\leq k(\log {p_{k}}-1)$

$k(\log {p_{k}}-0.1)\leq p_{k}+k\leq k\log {p_{k}}$

$p_{k+1}\leq p_{k}+k\leq k\log {p_{k}}$

$p_{k+1}-k\leq p_{k}\leq k\log {p_{k}}-k$

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

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 $e^{x}\geq 1+x$ therefore $\log(e^{x})\geq \log(1+x)$, so $x\geq \log(1+x)$.

Two related conjectures (see OEIS: A182514 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.