Remainder of a power tower under modulo $2013$

I have an expression like this: $$\left(\large 6000^{5999^{5998^{5997^{{\ldots^{1}}}}}}\right)\bmod 2013$$ Then which method should I use to solve it? Please provide the method not the answer. Editor’s Note: Note that this is a power tower with different values and not the same value as with general tetration. Also, don’t confuse tetration with exponentiation. Both are completely … Read more

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham’s number and before we can even start calculating $g_1$ which is: $g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of: $3 \uparrow\uparrow\uparrow 3$. I know it is a power tower of 3’s that is 7.6 trillion high (which would reach from the earth to sun … Read more

Fixed Point of xn+1=ixnx_{n+1}=i^{x_n} [duplicate]

This question already has answers here: Complex towers: iii… (2 answers) Closed 5 years ago. For x∈C, let f(x)=ix=exp(iπx), where i2=−1. Then find the fixed points for f. EDIT: Let for all n≥1 an=ii⋯i⏟n times My question is, does the sequence of tetrations {an}n≥1 converge to some complex number? If yes, then what is it? Answer … Read more

How iterated exponential \exp^{[\circ x]}(y)\exp^{[\circ x]}(y), y\neq 1y\neq 1, defined based on tetration?

Background: The tetration \begin{equation} ^xe = \exp^{[\circ x]}(1) = \underbrace{e^{e^{\cdot^{\cdot^e}}}}_{x \text{ times}} \end{equation} is well defined when x \in \mathbb{Z}. The extension of tetration to real height x \in \mathbb{R} can also be understood (though not unique). For instance ^xe \approx 1+x for -1 < x \leq 0, and this can be iterated to interpret … Read more

Integrating an Infinite exponent tower

∫10x2x2x…  dx=  ?                                                                                                                                           What I’ve tried so far: x2x2…def⏞=yx2y=y2yln(x)=ln(y) Now applying the chain rule: 2y⋅dydx⋅ln(2)⋅ln(x)+1x⋅2yln(y)=dydx⋅1y⟹dydx(2y⋅ln(x)⋅ln(2)−1y)=−1x2y⋅ln(y)⟹dydx=(−1x2y⋅ln(y)2y⋅ln(x)⋅ln(2)−1y) The problem that this looks like a pretty harsh equation to solve for dy and then substitute it back to the integral, with the fact that we need to find the value of x in terms of y. Any … Read more

Solving x1/x=y1/yx^{1/x}=y^{1/y} iteratively by power towers, convergence

Suppose we have no idea that x1/x=exp(1xlnx) or don’t know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. Then we can solve this equation the following way: x1/x=y1/y Denote y√y=z. x=zx=zzx=zzzzx==zzzz⋯ Really, we just do the following backwards recursion: x0=1 xk+1=zxk We know that the function f(x)=x1/x … Read more

Solution Verification – Integral of infinite power tower

$$\int x^{x^{\cdot^{\cdot^\cdot}}}dx=\int -\frac{W(-\ln(x))}{\ln(x)}dx$$ Let $u=-\frac{W(-\ln(x))}{\ln(x)}$, $\frac{du}{dx}=\frac{u^2}{(1-\ln(u))u^{\frac{1}{u}}}\Rightarrow dx=\frac{(1-\ln(u))u^{\frac{1}{u}}}{u^2}du$. This derivative is not trivial but I don’t have enough space to solve it here. I had a solution verification question were I solved this a while back Link for question. Many youtubers have also covered this. Using this information we can continue our integral: $$\int (1-\ln(u))u^{\frac{1}{u}-1}du$$ Let … Read more

Stronger statement : 6x≥e(ln2(x+1)ln(2)−ln(2))≥(x2+1)0.5^{6}x\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5

Let x>0 then prove or disprove that : xxxxxx≥e(ln2(x+1)ln(2)−ln(2))≥(x2+1)0.5 My attempt : Claim : Let 0<x≤1 and 0<a<1 and f(x)=ln(xxaaaa) then : f″(x)>0 Proof : f”(x)= 2 a^{a^{a^a}} x^{a^{a^{a^{a}}}-2} – x^{a^{a^{a^{a}}}-2} + (a^{a^{a^{a}}} – 1) a^{a^{a^{a}}} x^{a^{a^{a^{a}}}-2} \ln(x) It’s straightforward since 0.5\leq a^{a^{a^{a}}}\leq 1 So the function e^{f(x)} is log concave it conducts to the … Read more

Uniquely extended fractional iterations of \exp\exp

Let us define the following basic conditions for an iterated exponential function: \exp^1(x)=e^x\tag{$\forall x$} \exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$} I then pondered what sort of additional conditions could be applied. Using the useful inequality e^x-1\ge x, I considered adding the additional constraint: \exp^a(x)-a\ge\exp^b(x)-b\tag{$a\ge b$} which can be seen as a reasonable result of inductively applying the inequality. From … Read more

Simplifying a ‘fractal-like’ expression with tetration

Let f2(n)=2nn and let f3 be defined recursively as f3(n)=f2⋯f2⏟n times(n)=fn2(n). This will lead to tetration, but is it possible to write f3 in a closed formula, using the notation na for tetration (or even Knuth’s up-arrow notation)? I tried to write a simple code to see an emerging pattern, but simplifications seem somehow tricky. Also, … Read more