For questions about mathematical constants, that are "significantly interesting in some way".

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34 views

Why is it not known if Mill's constant is rational or irrational?

The following text appears in the Mill's constant definition at the Wikipedia: There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational (...
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0answers
88 views

Looking for a constant from complex analysis, ( and maybe other interesting constants) [closed]

Looking for a constant that is in complex analysis, it is about ratio of two types of sets in complex analysis. I have tried googling, wikipedia to no avail. I think I read about a constant ratios of ...
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0answers
132 views

Are my calculations of a new constant similar to Mill's constant based on $\lfloor A^{2^{n}}\rfloor$ and Bertrand's postulate correct?

As Wikipedia explains in number theory, Mills' constant is defined as: "The smallest positive real number $A$ such that the floor function of the double exponential function $\lfloor A^{3^{n}}\...
4
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3answers
174 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
16
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1answer
2k views

Why can't I find anyone who has discovered the (irrational) constant 1.29128…? [closed]

The constant is exactly $\sum_{n=1}^∞\frac{1}{n^n}$. Why does it seem that no one has written about it? Did I not search well enough? If so, what is the name for it? If not, it is not sufficiently "...
6
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1answer
98 views

How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$ \int_{0}^{\infty}\frac{1}{\Gamma(x)}dx $$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the Fransén-...
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2answers
31 views

Show that an entire function is a constant [duplicate]

Let $f$ be entire and suppose there is a constant $M>0$ such that $|f(z)|>M$ for all $z \in \mathbb C$. Prove that $f$ is constant. I think this has something to do with Liouville's theorem but ...
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1answer
141 views

How values of the constants are derived mathematically? [closed]

As said by Jan regarding constant value $\pi$ ,Imagine you have a circle and you are able to measure its circumference "c". Then, you can also find out what its diameter "d" is. When you divide ...
3
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1answer
89 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
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3answers
2k views

On the “Look-and-Say” sequence and Conway's constant

The look-and-say sequence starting with $S_1=1$ is, $$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$ If $L_n$ is the number of digits of the $n$th term then, $$\lim_{n\to\infty}...
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5k views

What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ...
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1answer
72 views

What is the value of $a + b + s + t$

I'm confused as to how to approach this. I've tried to foil it out. $$x^3(3x-1)=a+bx+sx^2+tx^3$$ The equation above is true for all values of $x$, where $a$, $b$, $s$, and $t$ are constants. What ...
0
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1answer
74 views

General solution of the differential equation: y' cot x + y = 2

I have to find the general solution of the differential equation:$ y$' $cot$ $x$ + $y$ = $2$. And determine the integration constant using the initial condition $y$(0) = $1$. Additionally presenting ...
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0answers
64 views

Ways to determine $\pi$ [duplicate]

I have read that it is possible to determine the value of a single digit, say the 874th of $\pi$. I know that it is a trascentental number, how is that possible? How many ways are there to determine ...
7
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1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
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1answer
35 views

Family functions of derivatives and extremum values. [closed]

Explore the family of functions $ f(x) = x^3 + kx + 1 $ where $ k $ is a real constant. How many and what type of local extrema are there? Your answer should depend on the value of $ k $, that is, ...
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2answers
76 views

Why square a constant when determining variance of a random variable?

If I want to calculate the sample variance such as below: Which becomes: $\left(\frac{1}{n}\right)^2 \cdot n(\sigma^2)= \frac{\sigma^2}{n} $... My question is WHY does it become $$\left(\frac{1}{n}...
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0answers
43 views

Why do we care about the Champernowne constant?

I was browsing code golf and I came across this challenge: http://codegolf.stackexchange.com/questions/68685/the-rien-number It caught my interest and I wanted to learn a bit more about the ...
2
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3answers
54 views

Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
0
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1answer
32 views

Problem with constants in Solving first order differential equation with perturbation

$$y'+\lambda \ y^4 +y=0$$ Where $\lambda$ is very small and $y(1)=-0.5$. I've tried to solve it by Substituting: $y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots$ but I had problems with the integration ...
14
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1answer
485 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
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1answer
24 views

Skipping integration constants

Here is simple decay equation – its initial conditions are being derived: $$ m(t) = Ce^{-kt}\\ m_0 = C^{-kt_0}\text{, which gives}\\ C = m_0e^{kt_0}\\ \text{After inserting $C$ to first equation}\\ m(...
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1answer
25 views

Finding Constants on a Differential Equation

Question (from my sample exercises in calculus): Find constants $A, B$ and $C$ such that the function $$ y=A\sin x+B\cos x $$ satisfies the differential equation $$ y''+y'-2y=\sin x. $$ I am ...
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1answer
65 views

How to prove the following formula is a constant?

For any $\rho\in\mathbb{R}^+$, prove that the following formula equals a constant: $$\dfrac{1}{\rho^2}{\int_{-\rho}^\rho x^2 e^{\left(\tfrac{\rho^2}{x^2-\rho^2}\right)}dx}\left({\int_{-\rho}^\rho e^{\...
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20 views

Universal Parabolic Constant

I recently learned of a constant that arises in parabolas, similar to that of $\pi$ for circles. Like $\pi$ being the ratio of the circumference of the circle to its diameter, this constant $\sqrt2+\...
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3answers
115 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
2
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2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
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0answers
49 views

Why only a handful of distinct mathematical constants appears in most applications?

Is there some theoretical reasoning behind the fact that the same numbers ($\pi$, $e$, $\gamma$, etc) appear again and again in every application in every field of Mathematics? Most people just ...
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126 views

Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. So for example $\frac{1}{1^1} + \frac{1}{2^1} + \frac{1}{3^2} +\frac{1}{4^2} + \...
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3answers
45 views

Can you have many integrations for a single function? (sorry if the terminology is wrong)

Forgive me for the amateur question (first time here), but it's bugging me! OK, for example, integrating 1/5x would yield what? ln(5x)/5 +C OR ln(x)/5 +C ? Both of these when differentiated form 1/(...
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62 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over 2}-{1+\...
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2answers
78 views

Find the limit $\lim_{h\to 0}\frac{ \sqrt{7(a+h)}-\sqrt{7a} }{h} $ in terms of $a$ .

Find the limit in terms of the constant $a$ : $$ \lim_{h \to 0}\frac{\sqrt{7(a+h)}-\sqrt{7a}}{h} $$ I have tried to solve this by multiplying the square roots to both sides but i simply can't solve it ...
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1answer
75 views

Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods? UPDATE Well now I know it should not. But $e^{-\gamma}$ should.
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1answer
39 views

Moment generating function of a constant

This might be trivial, but can you elaborate why moment generating function for a constant $c$ is $e^{cX}$, where $X$ is a random variable.
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1answer
22 views

Solve series with two constants and opposite exponents

How can I find the general formula for the sum of this series? $$ \sum_{i=0}^n a^ib^{n-i} $$ Where $a$ and $b$ are unrelated constants? I don't think you can split it into $ \sum_{i=0}^n a^i $ and ...
5
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1answer
86 views

Proving $\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$

How to prove that $$\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$$ Where $F_{n}$ is the Fibonacci Number.
0
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1answer
56 views

Sum of infinite series - Catalan constant

Why is this identity true? $$\sum_{k=1}^{\infty} \frac{sin(k\pi/2)}{k^2} = C$$ where $C$ is Catalan constant.
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1answer
477 views

MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper whose author has passed away. ...
4
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1answer
121 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that $$\...
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3answers
538 views

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as ...
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1answer
136 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
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2answers
65 views

Is that a known “constant”?

I noticed, using a calculator, that the following operation: $$Y = \sin(\cos(\tan(\log(n))))$$ for values of $n$ from $0.00000001$ to $99999999$, I obtain always quite the same number, id est $$Y = ...
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2answers
41 views

Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?

I believe that the sum $$\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this ...
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1answer
28 views

Taking derivatives of constants and variables

When implicitly differentiate a function, for example, $f(x)=(G)(x)$, where G is a constant, is it possible to differentiate it such that we can treat G as a variable? From my understanding it is ...
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43 views

Show that $\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$

Let $p(n)$ denote the probability that a randomly chosen one-to-one function $f:\lbrace 1,2,3...,n \rbrace \rightarrow \lbrace 1,2,3...,n \rbrace$ has a least one fixed point. (least one integer $k$ ...
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0answers
56 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
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23 views

Finding the constant in a joint density function

I have been having a bit of difficulty with the following example, whenever i integrate the terms vanish. Any help is appreciated! The joint density function of X and Y is given by $$f(x , y) = $$\...
0
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1answer
44 views

Find the constants to make it continuous.

I am having trouble starting this question. Find the constants (a, b or c) to make the function continuous. f(x)= { ...
3
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1answer
52 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...