# Questions tagged [square-numbers]

This tag is for questions involving square numbers. A non-negative integer $n$ is called a square number if $n = k^2$ for some integer $k$. Consider using with the [elementary-number-theory] or [number-theory] tags.

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### $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
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### Is there $n \geq 2$ such that $1^1 + 2^2 + \dots + n^n$ is a perfect square?

Define $S_n = 1^1 + 2^2 + 3^3 + \dots + (n - 1)^{n - 1} + n^n$. It is clear that $S_1 = 1^1 = 1$ is a perfect square. However, I have found no others so far. Question: Does there exist $n \geq 2$ ...
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### Can the sum of powers of the first primes be a square?

Let $p$ be a prime and $u\ge 1$ be a positive integer. Define \begin{align} S(p,u) &:= \sum_{q\text{ prime, }q \le p} q^u \\ &= 2^u+3^u+\cdots +p^u\end{align} I wonder whether $S(p,u)$ ...
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### What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
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### Prove that every integer greater than 1 is sum of square and squarefree.

I tried to put some sieve method on this. Here, we denote the classic squarefree sieve function $\sum_{d^2|a}\mu(d)$ (notice that if $a$ is squarefree, the value is 1, and if not squarefree, the value ...
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### Is there any perfect square in the sequence $12,123,1234,12345,…$?

Let $x_n$ be the number constructed by concatenating the first $n$ positive integers. Is there any perfect square in the sequence $(x_n)_{n ā„ 2}$ ? This is OEIS A007908. This is the sequence of ...
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### When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ ...
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### is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere?

is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere? just to keep track $\frac{a}{b}$ is the simplest form of $x$ when $x$ is rational. $\frac{c}{d}$ is the ...
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### a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
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### Is $p^2+q^2+r^2=3^k$ with primes $p,q,r$ solvable for every odd positive integer $k\ge 3\$?

For the positive odd integers $3\le k\le 25$, the equation $$p^2+q^2+r^2=3^k$$ with primes $p,q,r$ is solvable. Here is one solution for every exponent , calculated with PARI/GP : ...
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### Are there other values of $n$ that generate $p^2$?

I found a pattern that looks quite interesting. \begin{align} 2(4 + 2) + 13^3 &= 47^2 \\ 2(4 +5) + 7^3 &= 19^2 \\ 2(4 + 8) + 1^3 &= 5^2.\end{align} It seems at first that if $p$ is ...
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### Square Roots: Variables with Exponents.

Alright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?
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### Are all solutions of $f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
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### Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...
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### A bound on the squares of primes

If $p_n$ is the $n$th prime, what is the best known upper bound on $m$ such that $p_n \cdot p_{m+n} < p_{n+1}^2$?
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### When does a certain number is a perfect square

I've the following number: $$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$ Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
I have been looking for an algebraic way to solve for 2 squares when given a sum, but I got nothing so far. For example: $$y^2 + x^2 = 9797$$ I thought that the solution would have to do something ...