# Questions tagged [square-numbers]

A non-negative integer $n$ is a square number if $n = k^2$ for some integer $k$.

816 questions
2answers
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### Find $\int \frac{1}{\sqrt{-x^2-6x+40}}dx$ using completing the square?

I am not sure how to find the integral by completing the square here since it's inside of a square root. I am practicing with Khan Academy, and I have four choices for answers, all of which include ...
2answers
51 views

### What are the possible solutions for the diophantine equation $4x^2-3y^2=1$ and is there a general formula?

Assuming that $a = x^2$ and $b = y^2$, i converted this equation to a linear diophantine equation for sake of convenience: $$4a - 3b = 1$$ where after calculating a particular solution (like $(1, 1)$...
1answer
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### Weird phenomenon with the perfect squares of numbers under 14.

In math class (algebra 1), a classmate of mine realized this weird thing when asked the square or 21. In her head, knowing that $12^2$ = 144, she said 12 flipped is 21 so 144 flipped is 441, which is, ...
4answers
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### Playing with squares

Extending from particular examples I've found that $$n^2=\sum_{i=1}^{i=n-1} 2\, i+n$$ this is that for any square of side $n$ the area can be calculated in a simple way. Example For a square of ...
2answers
39 views

### If $a^2 = b^2$ then which values $a$ and $b$ are constrained to be? [closed]

I've the following subset of $\mathbb{R}^3$: $$Y= \{(a, b,c)^T | a^2=b^2\} \subset \mathbb{R}^3$$ How can I embed the condition $a^2=b^2$ into the vector? That is, what can I say about $a$ and $b$?...
1answer
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### What is the difference between, a “square” and a “perfect-square”, number?

Is, "36", a perfect square? I know that, "4" is a perfect square. Similarly, "1","9","25", are "perfect-square"s.
2answers
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### Perfect square root recurrence

Spent some time trying to find some recurrence for determine bigger than current perfect square but unsuccessful. For example: current 121 and next 144. Who is next after 144? Can someone help me to ...
0answers
12 views

### Equality regarding the square of the sample mean

Given that $X_1,...,X_n$ is an i.i.d sample and its sample mean is $\overline X_n$, I have to prove the following equation: \begin{equation*} \frac{n-1}{n} \sum_{i=1}^n(\overline{X}_{n-1,i}^2 ...
2answers
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### What is the smallest $n$ such that $\frac{n(n+1)(2n+1)}{6}$ is a square number? [closed]

Question : Find the smallest natural number $n>1$ such that $\sum_{k=1}^{n}k^2$ is a square number Recall that : $\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$
2answers
732 views

### If $u_1=1$ and $u_{n+1} = n+\sum_{k=1}^n u_k^2$, then $u_n$ is never a square.

Cute problem I saw on quora: If $$u_1 = 1,\qquad u_{n+1} = n+\sum_{k=1}^n u_k^2$$ show that, for $n \ge 2$, $u_n$ is never a square. \begin{align} n&=1:& u_2 &= 1+1 = 2\\ n&=2:& ...
1answer
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### Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$. Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$. Is that ...
0answers
59 views

### Number of ways of proving that a number given in algebraic form is a perfect square or is not.

I've listed a few ways by which it can be proved. Please correct me if any of these are wrong. Are there other possible ways which I'm missing out? If an even number is a perfect square, it must be ...
1answer
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### When are numbers of the form $m^2+9k^2\pm k$ perfect squares?

In pursuing a question of personal interest, I need to characterize solutions to the equation $m^2+9k^2\pm k=c^2$; in other words, I want to know when numbers of the form $m^2+9k^2\pm k$ are perfect ...
1answer
65 views

### Finding a perfect square within an interval

I have a function $f(x) = 4x^2 + 4ax + b$ where $a$ and $b$ are both even numbers. I also have an interval within which there are only two integers $x$ that will result in a perfect square. $a$, $b$, ...
1answer
39 views

### Diophantine equation $dx^2-y^2=d-1$ for non-square $d$

For $d=2$ or $3$ the continuous fractions for $\sqrt 2$ and $\sqrt 3$ help (at least partially), but for $d\ge 5$, this doesn't seem to work. Is there any known solution or technique to solve this ...
1answer
65 views

### How to solve Shonk Sequences?

A Shonk sequence is a sequence of positive integers in which each term after the first is greater than the previous term, and the product of all the terms is a perfect square For example: 2, 6, 27 ...
5answers
2k views

### Why does this pattern in powers happen? [duplicate]

Hopefully this not to frivolous to ask, I have wondered it for years. There is a strange pattern in the differences between various power. For example if you square the numbers from one to ten the ...
3answers
38 views

### Quadratic square values

Find the value(s) of positive integer $n$ such that $n² + 19n + 48$ is a perfect square. I factorised it to $(n+3)(n+16)$, but that gives negative integer answers $-3$ and $-16$. What do I do?
1answer
34 views

### Square of an octal number

How to find the square of an octal number. For example what will be the square of 23. It will not be 529 because octal number system does not have digit 7
1answer
34 views

### Showing that the congruum is divisible by 24

Let $a,b,c \in \mathbb{N}$ be four natural numbers satisfying $b^2-a^2=c^2-b^2$. That is, $a^2, b^2, c^2$ are three successive squares in an arithmetic progression. Show that $24$ divides $b^2-a^2$. ...
0answers
37 views

### Elementary Proof on Perfect Squares.

The Proof I'm working on is: If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational.\ The farthest I've gotten is by proving by contradiction, assuming that $\sqrt{r}$ is ...
1answer
63 views

### Prime numbers & perfect squares

Find all prime numbers such that $2p^4-p^2+16$ is a perfect square. $2p^4-p^2+16=n^2$ $16-n^2=p^2-2p^4$ $(4-n)(4+n)=p^2(1-2p^2)$ What should I do next?
1answer
52 views

### Perfect squares and divisor

Let $n$ be a positive integer and let $d$ be a positive divisor of $2n^2$. Prove that $n^2+d$ is not a perfect square. My working: $d \mid 2n^2$ Let $d \cdot k=2n^2 \implies d=\dfrac {2n^2}k$ ...
1answer
48 views

### The units digit of a perfect square is 6. What are the possible values of the tens digit? [closed]

I know the answer to this already: the possible values of the tens digit are 1, 3, 5, 7, and 9. But I don't know how to prove it, can someone help please? Thanks!
1answer
54 views

### Perfect squares with two variables.

Find all positive integers m, n such that $6^m + 2^n + 2$ is a perfect square. I've tried keeping a constant value of m and finding out n. Eg: $m=1, n=0$ $m=1, n=3$
2answers
137 views

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### If $q$ is a Fermat prime, is $\sigma(q^k)/2$ a square if $k \equiv 1 \pmod 4$?

In what follows, let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number of the form $M = 2^{2^m} + 1$ is called a Fermat number. If in addition $M$ is prime, then $M$ is ...
1answer
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1answer
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### Finding solutions in $\mathbb{Z}_{+}$ [closed]

Find all the triplets of positive integers $(a, b, c)$ such that: $a^2+b+3=(b^2-c^2)^2$
3answers
40 views

### Square numbers related to square root

Given whole numbers $a$, $b$, $c$ satisfying $\sqrt{a}+\sqrt{b}+\sqrt{c}$ are also whole numbers. Prove that $a$, $b$, $c$ are square numbers.
1answer
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2answers
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### Show that if $a,b,c\in \mathbb{N}$ and ${a^2+b^2+c^2}\over{abc+1}$ is an integer it is the sum of two nonzero squares

I was reading about the fascinating problem in the IMO ($1988$ #$6$) that asks: Let $a$ and $b$ be positive integers such that $(ab+1) | (a^2+b^2)$. Show that ${a^2+b^2}\over{ab+1}$ is a perfect ...