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24
votes
3answers
1k views
39
votes
4answers
2k views
Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?
Problem:
Find $x$ in
$$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$
Trick:
$x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so,
$x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and,
...
5
votes
4answers
1k views
Non-integer powers of negative numbers
Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically:
Can we define fractional powers such as (-2)^-1.5?
Can we define ...
2
votes
5answers
1k views
Why is two to the power of zero equal to binary one?
Probably a simple question and possibly not asked very well. What I want to know is..
In binary, a decimal value of 1 is also 1.
It can be expressed as $x = 1 \times 2^0$
Question:
Why is two to ...
86
votes
3answers
3k views
Is 2048 the highest power of 2 with all even digits (base ten)?
I have a friend who turned 32 recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being 32 was pretty good since not only is it even, it also has no odd factors. ...
8
votes
6answers
370 views
Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$
I'm so puzzled about this:
$$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$
Why isn't $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that ...
4
votes
1answer
503 views
Prove variant of triangle inequality containing p-th power for 0 < p < 1
Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while:
$\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
2
votes
4answers
3k views
Modulo arithmetic with big numbers?
I need to calculate $3781^{23947} \pmod{31847}$. Does anyone know how to do it? I can't do it with regular calculators since the numbers are too big. Is there any other easy solutions?
Thanks
1
vote
2answers
67 views
Calculating new vector positions
I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
6
votes
3answers
476 views
What we can say about $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$?
Problem:
How we can strictly prove $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ can't be 2?
Can $-\sqrt{2}^{-\sqrt{2}^{-\sqrt{2}^\ldots}}$ have the value expressed by complex numbers? (See below, in ...
14
votes
2answers
255 views
Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime
My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
4
votes
1answer
147 views
Are limits on exponents in moduli possible?
Suppose I show that:
$$x^{f(z)/g(z)} = y \pmod{4}$$
is impossible for some given positive integers $x$ and $y$, where,
\begin{align*}
f(z) &= \phi(4) k_1(z) + 1 \\
&= 2 k_1(z) + 1\\
g(z) ...
4
votes
1answer
486 views
Algorithm for computing square root of a perfect square integer?
My question is the following:
Is there a polytime non-numerical
algorithm for computing square root of
perfect square integers?
The more elementary the algorithm is, the better!
EDIT:
...
2
votes
5answers
68 views
Proof for power functions
Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$
I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
8
votes
1answer
332 views
Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$?
$x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
5
votes
4answers
2k views
Negative 1 to the power of Infinity
Can anyone explain me what the result of $$\lim_{n\rightarrow\infty} (-1)^n$$ is and the reason?
4
votes
1answer
238 views
Comparing Powers of Different Bases
How can I know if one power is bigger than the other when the bases are different?
For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
3
votes
2answers
224 views
2
votes
5answers
523 views
What is the value of $2^{3000}$
What is the value of $2^{3000}$?
How to calculate it using a programming language like C#?
