# Questions tagged [exponentiation]

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

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### 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 $\exp(aD)f(x)=f(x+a)$?
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### How many solutions to each of the equations $2^x+3^y=5^z$, $2^x+5^y=3^z$, $3^x+5^y=2^z?$

Let $x,y,z\in\Bbb{N}$ How many total solutions are there to each of the three distinct equations below? $$2^x+3^y=5^z \tag 1$$ $$2^x+5^y=3^z \tag 2$$ $$3^x+5^y=2^z \tag 3$$ I found 3 solutions to ...
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### Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that "....
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### Is there a notation for exponentiation analog to capital-sigma notation Ī£ for addition and capital pi Ī  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 (...
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### Can this type of limit even be evaluated?

This is going to be a long question; please bear with me. We are familiar with the notations $\Sigma^{n}_{k=0} a_k$ and $\Pi^{n}_{k=0}a_k$ for the sum and product of the finite sequence $\{a_n\}$. I'...
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