# 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.).

248 questions
Filter by
Sorted by
Tagged with
4answers
44 views

1answer
300 views

### Solutions to $A^N+B^N=C^N \pm 1$

Is there a solution to $A^N+B^N=C^N \pm 1$ where $A,B,C,N\in\Bbb{N}$, such that $A,B,C > 1,N \geq 4$ and $gcd(A,B)=gcd(B,C)=gcd(A,C)=1$? This question was inspired by this one: $A^X+B^Y=C^Z\pm 1$ ...
0answers
46 views

### Power Tower and Finite Tetration

Let $$\underbrace{a^{a^{.^{.^{.}}}}}_{2020} =2020$$ How can I solve for $a$? Had this been an infinite power tower, I could have used the self-replication. But the finite number of terms (i.e. $2020$ ...
1answer
137 views

### When do we have $(x y)^2 = x^2 y^2$?

I just started thinking about algebra so this might be a trivial question. Anyway, Under what conditions do we have $$(x y)^2 = x^2 y^2$$ ? Does it need to be a group ? Or a groupoid ? Or a monoid ? ...
0answers
74 views

1answer
130 views

### What is the $n^\text{th}$ perfect power, $P(n)$?

Let $P(n)$ denotes the $n^\text{th}$ perfect power of natural numbers (in ascending order without repetition). So, $P(1)=1, P(2)=4, P(3)=8, P(4)=9, P(5)=16, P(6)=25, P(7)=27, P(8)=32, \dots$. Is ...
0answers
77 views

### Conjecture That The Only Nearly Perfect Numbers Are The Powers Of 2

Call a natural number $n$ “flawed” if its aliquot sum (the sum of its factors less than itself) is $n-1$. Show that the set of flawed numbers is equivalent to the set of perfect powers of $2$. ...
3answers
724 views

### Pythagorean triple with hypotenuse a power of $2$

Is there a Pythagorean triple whose largest element is a power of 2? That is: are there solutions to $a^2 + b^2 = 2^{2k}$ in the positive integers?
3answers
601 views

2answers
318 views

### How many zeros in the decimal representation of $5^n$?

I'm curious about some properties of the powers of 5 $$5^2=25,\quad5^3=125,\quad 5^4=625,\quad 5^5=3125,\quad ...$$ Is it true that at least $50$% of the digits in the decimal representation of $5^n$ ...