Questions about exponentiation

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8
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79 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
6
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179 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
6
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184 views

The exponential map

I'm following a course about riemannian geometry, and I was fascinated with the exponential map. I was wondering what the reason of this name is... is there any relationship with the real and complex ...
5
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0answers
198 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
4
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126 views

Convergence in Growth/Decay of Sum Odd and Sum Even Exponentiated Terms

where $n \geq 2 $ Given that a function containing an odd number of exponentiated terms as follows $$\left(\frac1{n}\right)^{\left(\frac1{n+1}\right)^{.^{.^{.^{\left(\frac1{n+2m+1}\right)}}}}} $$ ...
3
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0answers
81 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
3
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0answers
113 views

Bernoulli formula

The sum: $$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$ Can be calculated by this formula, called the "Bernoulli formula" in wikipedia $$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k ...
3
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0answers
73 views

Given an integer, how can I detect the nearest integer perfect power efficiently?

If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N? In other words, the perfect power the distance between N and which is less than the ...
3
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0answers
243 views

What is the Equivalent Form of Tetration to the Exponential $n^{1/n}$?

I've been working on a project for a wiki that I'm a member of. It is the Sequence of the Day for September 2. You can see my progress at ...
2
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0answers
40 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
2
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0answers
37 views

Exponential equations solving methods?

Do you have an idea or general method to solve the following equation?: $$a^{\alpha x}+b^{\beta x} = c^{\gamma x}+ d^{\delta x}$$ when $a,b,c,d$ aren't zero, and $\alpha, \beta, \gamma, \delta$ are ...
2
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0answers
79 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
2
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0answers
71 views

Prove $a^b=\underbrace{a \times a \times \cdots \times a}_\mathrm{b\ times}$

Starting with the definition $a^b = \exp(b \ln a)$, where $\exp$ is defined by its Maclaurin series and $\ln$ is defined by the integral $$\ln x = \int_1^x \frac{dt}{t}$$ how can we show that ...
2
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0answers
76 views

Sums of powers.

Here's the problem: Show that $19^{19}$ is not the sum of a fourth power and a positive or negative cube. I'm just not really sure how to start approaching this problem. Does anybody have any ...
2
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0answers
138 views

Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?

Solving $x$ for $y = x^x$ using Lambert W function is clear enough thanks to this handy answer, but as I'm using the solution in a network support document I need it in a form that can be solved on a ...
2
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0answers
89 views

Is $x^{\frac 12}$ the same as sqrt(x)

maybe this question is very simple and clear and trivial to everbody. but right now i'm not sure. the equotation $$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right? the square ...
2
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0answers
37 views

Can we define root extraction using Peano Arithmetic?

I've been playing with Peano Arithmetic and I've got multiplication, division, exponentiation, and logarithms. I can't figure out root extraction but I have a stab at it. Exponentiation: $a^0 = 1, ...
2
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0answers
68 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
2
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0answers
234 views

Exponent p-value generated in Excel

Excel gave me a p-value of 1.44909E-09 Notice is does not say .09 but 09 This is confusing me, I am trying to analyze my data but am stuck at this point. If it were E-9 it could be ...
2
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0answers
170 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
2
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0answers
418 views

upper bound of exponential function

I am looking for a tight upper bound of exponential function (or sum of exponential functions): $e^x<f(x)$ when $x<0$ or $\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$ Thanks a ...
2
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0answers
354 views

exp(ab) decomposition

How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
2
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0answers
114 views

Determining maximum value of $a^b\bmod N$ when $\gcd(a,b)$ is known

Suppose we know the greatest common divisor, $\gcd(A,B)$, of two numbers $A$ and $B$. Is there a way that we can find the maximum value of $a^b \bmod N$ where $N$ is any number? We have a finite ...
2
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0answers
86 views

Finding the value of $x$ for an equation

If we have the expression $a=x^{c\cdot x+1}$ where the values of $a,c$ are known, how can we find the value of $x$? I tried using log but it yields: $x = a ^ {(1/x)/(c-1/x)}$ from which I can't find ...
1
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0answers
61 views

Rising/Falling Powers, Summation

1) Show that $$(-n)^{\bar p} = (-1)^n n^\underline{p}$$ (original screenshot) 2) Evaluate the sum $$\sum_{a\le n\lt b}n^{\bar p}$$ (original screenshot) Thoughts regarding question 1: I've ...
1
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0answers
32 views

Algebra of Exponential and Log Functions

This question may have a simple answer or a very complex one, but I am interested in what the reasons are for logarithms and exponential functions having the properties they have. To my knowledge ...
1
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0answers
23 views

Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are: It mentions that Lieb's thoerem "accomplishes ...
1
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0answers
19 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
1
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0answers
24 views

Math equations of electron scattering

I'm trying to figure out the missing step here, in a problem about X-ray crystallography. I am referring to the attached image: In the image, A= electron density, Z= distance traveled, λ= X-ray ...
1
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0answers
46 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
1
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0answers
55 views

Algorithm for finding power

I has been searching for a high precision library in PHP to do calculations like $$232323232323^{121212.2232323232}$$ etc (ie, with very large numbers, including decimals), but failed to get any. ...
1
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0answers
524 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
1
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0answers
46 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
1
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0answers
79 views

Generalisation of Lambert W function?

I want to solve an equation of the form: $\exp(C / x) - 1 = D / (x + a)$ This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ...
1
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0answers
56 views

How to solve for the matrix in a set of equations involving the matrix exponential?

I was wondering how to solve the following problem (in a least-squares sense): $$ \mathbf{y}_1 = e^{Ax_1} \mathbf{y}_0 \\ \mathbf{y}_2 = e^{Ax_2} \mathbf{y}_0 \\ \vdots\\ \mathbf{y}_n = e^{Ax_n} ...
1
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0answers
176 views

jacobian involving SO(3) exponential map: $\log(R * \exp(m))$

I would like to compute the 3x3 Jacobian of $$ \log(R * \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
1
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0answers
69 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...
1
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0answers
153 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
1
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0answers
82 views

Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
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0answers
117 views

Solving for $x$ in $y=x^x(\ln x + 1)$ (Lambert W?)

I made a bunch of problems exercising the Lambert W-function in the solution, because I like to exercise to new concepts that I learn about. One that I came up with was rearranging $y = x^x(\ln x + ...
1
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0answers
40 views

Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
1
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0answers
43 views

Is an equation of the following form solvable?

Is it possible to solve for $x$ which satisfies the equation $$d=(a\exp(bx)+\exp(cx))x^2$$ where $a,b,c,d$ are given constants? It looks quite horrible... Many thanks!
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0answers
96 views

How to solve these exponential equations for D?

I'm curious if this is even possible to solve for D. D is the only variable, x, y, z, and w are all constants, and e is the mathematical constant e. $$ (\frac{x+yD^{2}}{zD})^{\sqrt{2D}} = ...
1
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0answers
128 views

Notation for Cartesian power, an oddity in the international standard

The Cartesian product of $A$ with itself $n$ times is normally denoted using superscript notation $A^n$, and this what ISO 31-11 defined as standard. However, ISO 31-11 has been superseded by ISO ...
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0answers
135 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
1
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0answers
150 views

Solve $x^x=2x$ for $x$, s.t. $x\in\mathbb{C}$

Solve $x^x=2x$ for $x$, such that $x\in\mathbb{C}$. I'm not sure if the question has a closed form solution.
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0answers
101 views

Collaborative modular exponentiation

EDIT: Rephrased. I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$. I also know $b$, $R$ and $N_2$ (but not its factors). I want to know whether there is ...
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0answers
83 views

Integer exponentiation algorithm for the special case $3^n$

Is there any known integer exponentiation algorithm to compute $x^y$ for the special case $x = 3$ which is faster than the general case algorithm found in [1], section 4.6.3? [1] D. E. Knuth, The Art ...
1
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0answers
182 views

What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
1
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0answers
76 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...