Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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11
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0answers
125 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
votes
0answers
215 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 ...
5
votes
0answers
88 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
5
votes
0answers
118 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 ...
5
votes
0answers
244 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{^} ...
5
votes
0answers
217 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 ...
4
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0answers
59 views

Why can $x^0$ sometimes be simplified to 1 even when x can equal 0?

For example, the Taylor series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. It seems like it should be indeterminate or undefined at $x=0$, since the first term would contain $0^0$, but it's not ...
4
votes
0answers
92 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
4
votes
0answers
79 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
4
votes
0answers
264 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 ...
4
votes
0answers
145 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
votes
0answers
56 views

Solution to following functional equation

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
3
votes
0answers
57 views

Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation ...
3
votes
0answers
34 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
3
votes
0answers
136 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 ...
2
votes
0answers
35 views

How to reduce exponentiation expressions?

It is a simple question but I am afraid of its simplicity. Is that correct : $2^{30}+2^{30}+2^{30}+2^{30} = 2^{30}(1 + 1 + 1 + 1) = (2^{30})\cdot 4 = 2^{30}\cdot2^2 = 2^{32}$? I am doing complex ...
2
votes
0answers
30 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
2
votes
0answers
32 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
2
votes
0answers
110 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
2
votes
0answers
212 views

Find the value of a^b?

Given a series as f(n)=1^1 * 2^2 * 3^3 * ......n^n since n can be very large Find the value of f(n)/f(r)*f(n-r) and output it modulo m where m is any prime. Now My approach is f(n)=1^1%m * ...
2
votes
0answers
57 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...
2
votes
0answers
52 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
2
votes
0answers
88 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
2
votes
0answers
41 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
2
votes
0answers
79 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
2
votes
0answers
23 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
2
votes
0answers
53 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
votes
0answers
55 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
votes
0answers
83 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
votes
0answers
90 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
votes
0answers
218 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
votes
0answers
92 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
votes
0answers
58 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
votes
0answers
110 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
votes
0answers
1k 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
votes
0answers
248 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
votes
0answers
511 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
votes
0answers
131 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
votes
0answers
88 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
vote
0answers
31 views

A simple question on the matrix exponential

Probably a trivial question. Given two random matrices $A, B$ such that $\left\langle \left[A,B\right]\right\rangle =0$, namely only the (element-wise) mean of the commutator is zero, can I say that ...
1
vote
0answers
43 views

series of powers of integer powers

Given two real positive numbers $a,b\in(0,\infty)$ and a series of natural integers $n=1,2,3,\dots$, is there any known formula to apply in order to calculate the series $$s(n)=a^{b^n}?$$ My goal is ...
1
vote
0answers
14 views

Evaluating Expressions with Integer Exponents

Simplify the expression by writing as a single power and then evaluate for a=-1,b=-2 and c=3. $$(b^{3}a^{4})^{2} \times (a^{3}c)^{3} \over ac^{3}$$ Here is what I did: ...
1
vote
0answers
20 views

Help with repeated squaring

I'm having trouble figuring out how to use repeated squaring to figure out 289^377 mod 589. I've seen other websites break the exponent down into (1 + 4 + 16 ... ), but I'm not sure when to do that.
1
vote
0answers
38 views

Raising numbers with powers

So the question is as follows: Let $f(x) = \int_{0}^x \frac{x}{2\sqrt{t}}dt$. Suppose $f(f(f(...f(f(a))...)))$ (done $2013$ times) $= 2^{2013}$. Find the real-valued solution of $a$ Now, for my ...
1
vote
0answers
28 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
1
vote
0answers
66 views

Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...
1
vote
0answers
64 views

Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + ...
1
vote
0answers
19 views

Evaluating Exponents

Answer choices: 22 15 10 358 None of the above This is my solution:
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0answers
15 views

Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
1
vote
0answers
52 views

A variant of factorial

Given the definition of a function f as f(n)=1^1 * 2^2 * 3^3 * ... * (n-1)^(n-1) * n^n. Another function g is defined as g(n,r)=f(n)/(f(r)*f(n-r)) Given an n,r,m we are to output g(n,r)%m where m is ...