# Questions tagged [perfect-powers]

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. This tag should only be used when having in mind an arbitrary perfect power (as opposed to a specific one, like a perfect square, a perfect cube, etc.).

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### Techniques for solving an equation $f(x)=y^n$ over the integers?

Despite of the negative result of Hilbert's tenth problem, diophantine equations of the form $$f(x)=y^n$$ can often be fully solved. Here, $f(x)$ is a polynomial with integer coefficients and $n>1$ ...
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### Perfect powers nearest to factorials

Suppose, $n\ge 8$ is an integer. Let $s$ be the smallest non-negative integer such that $n!-s$ is a perfect power. Let $t$ be the smallest non-negative integer such that $n!+t$ is a perfect power. ...
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### Is there another perfect power?

This is related to this question : Can we conclude $n=p-1$? For which positive integers $\ n\$ is $$(2n)!+n!+1$$ a perfect power ? For $\ n=1\$ and $\ n = 2\$ , we have a perfect power and there is ...
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### Closest sums of pairs of reciprocals of prime numbers

This question is related to an upper bound for a problem from my five-year-old question. Let $n\ge 3$ be a natural number and $A_n$ be a set consisting of $1$ and first $n-1$ prime numbers. We want to ...
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### How can we prove that $3t$ cannot be a perfect cube for any integer $t$ except 9?

If $t \in \mathbb{Z}$ then prove that $3t$ can never be a perfect cube except for $t=9$. How can we prove things like these? I’m pretty new to Number-Theory and I find it difficult to prove things ...
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Given a positive integer $n> 2$. Prove that there exists a natural number $K$ such that for all integers $k \ge K$ on the open interval $\big({{k} ^{n}}, \ {{(k + 1)} ^{n}}\big)$ there are $... 0answers 68 views ### How to find the sum of the remainders of a number modulo all the perfect powers less than it? Perfect powers are numbers of the form$ x^y $where$x\geq1$and$y>1$. For example, perfect powers are$\{1, 4, 8, 9, 16, 25, 27, 32, \cdots \}$. If$n$is a natural number, then we denote by$P(...
The perfect power of an integer $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ are $p_1^2, p_1^3\cdots p_1^{\alpha_1},p_2^2, p_2^3 \cdots p_2^{\alpha_2},(p_1 p_2)^2, (p_2 p_3)^2\cdots$ ...