# 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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### is 0 a perfect square

Based on my research, I found that there are many arguments about this statement, the main factor is the true definition of perfect square. Some said they are the squares of the whole numbers, but ...
61 views

### Find all pairs of primes $p, q$ such that 3$p^2q+16pq^2$ equals to square of natural number [closed]

Find all pairs of prime numbers $p, q$ such that 3$p^2q+16pq^2$ equals to square of natural number My attempt: I've been trying to calculate equation through square root but now I'm stuck, please help
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### Show that $4x^2-yz$ is a perfect square

Here is my problem. $A=xy+yz+zx$, where $x,y,z\in\mathbb{Z}$. It is known that if we add $1$ to $x$, and subtract $2$ from both $y$ and $z$, the value $A$ won't change. Prove that $-A$ is a square of ...
4k views

### Can any number of squares sum to a square?

Suppose $$a^2 = \sum_{i=1}^k b_i^2$$ where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct). Can any positive integer be the value of $k$? The reason I am ...
183 views

### Deciding if a number is a square in $\Bbb Z/n\Bbb Z$

I am looking for a systematic way of deciding if a given number is a square in $\Bbb Z/n\Bbb Z$. E.g. is $89$ a square in $\Bbb Z/n\Bbb Z$ for $n\in \{25,33,49\}$? Brute-forcing it would take too ...
56 views

### Prove that $\sqrt{n^2 + 1}$ is not an integer [closed]

Prove that $\sqrt{n^2 + 1}$ is not an integer The title sums it all... I have tried to prove it for the past hour but I'm just stuck... Tried using induction and assuming the opposite. The logic is ...
341 views

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### Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number ...
53 views

### Generalizing $\,r(n^2) = r(n)^2,\,$ for $\,r(n) :=$ reverse the digits of $n$

I'm assuming this theorem was found by someone else before, but I found this relationship between square numbers of 3 digits or less. The theorem is this: If you reverse the digits in a square number, ...
51 views

### Establishing infinitely many primes of the form $4k+1$.

Establish that there are infinitely many primes of the form $4k+1$. I was studying primitive roots, and it was recently proven that the odd prime divisors $n^2 +1$ are all of the form of $4k+1$. A ...
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### Square Triangle numbers and Pell's equation

So i have just started to study number theory and i was asked this question. Now i tried to search online and i found out pell equation can used to solve this question. Now in an online video i saw ...
621 views

### Is it possible that $2^{2A}+2^{2B}$ is a square number?

Let A and B be two positive integers greater than $0$. Is it possible that $2^{2A}+2^{2B}$ is a square number? I am having trouble with this exercise because I get the feeling the answer is no, but I ...
240 views

### Find all integers $m,\ n$ such that $m^2+4n$ and $n^2+4m$ are both squares. [closed]

Find all integers $m,\ n$ such that both $m^2+4n$ and $n^2+4m$ are perfect squares. I cannot solve this, except the cases when $m=n$.
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### Finding all positive integers $n$ such that $\{n,n+1,n+2,n+3,n+4,n+5\}$ partitions into two subsets such that the products of elements are the same [closed]

Question from 12th IMO: Find all positive integers $n\in\mathbb Z^+$ such that the set $\{n, n + 1, n + 2, n + 3, n + 4, n + 5\}$ can be split into two disjoint subsets such that the products of ...
103 views

### How many squares in a three-dimensional $n \times n \times n$ cartesian grid?

This brings the classical question to three dimensions. Given a three-dimensional Cartesian grid of $n \times n \times n$ points (that is $(n-1) \times (n-1) \times (n-1)$ unit cubes), how many ...