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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2answers
85 views

Prove that $3(p^2)+61$ is a perfect square only for $p = 1$

It is too be proved that $3(p^2)+61$ is a perfect square only for $$p=1 ; p=N$$ The question arises from a problem in arithmatic progression with a certain soft constraint of natural numbers. At the ...
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1answer
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$m$ and $n$ are odd numbers such that $|m^2 - n^2 + 1| \mid (n^2 - 1)$. Prove that $|m^2 - n^2 + 1|$ is a square number.

$m$ and $n$ are odd numbers such that $$\large |m^2 - n^2 + 1| \mid (n^2 - 1)$$. Prove that $|m^2 - n^2 + 1|$ is a square number. I have provided the solution below, but I am uncertain about the ...
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Knowing that $m$ and $n$ are positive integers satisfying $mn \mid m^2 + n^2 + m$, prove that $m$ is a square number.

Knowing that $m$ and $n$ are positive integers satisfying $$\large mn \mid m^2 + n^2 + m$$, prove that $m$ is a square number. We have that $mn \mid m^2 + n^2 + m \implies mn \mid (m^2 + n^2 + m)(n + ...
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2answers
91 views

Regarding perfect squares-2 [closed]

How to find all the positive integers $n$ such that $20n^2 -12n + 1$ is a perfect square?
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3answers
170 views

Regarding perfect squares

Is there any positive integer $n > 2$ such that $(n - 1)(5n - 1)$ is a perfect square? It is observed that $(n - 1)(5n - 1)$ is of the form $4k$ or $4k+ 1$. Affirmative answers were given by Pspl ...
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2answers
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What are rationalizing factors?

I don't know what rationalizing factors are and I need help finding $\sqrt[5]{a^2b^3c^4}$ There are some options A) $\sqrt[5]{a^3b^2c}$ B) $\sqrt[4]{a^3b^2c}$ C) $\sqrt[3]{a^3b^2c}$ D) $\sqrt{a^...
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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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1answer
48 views

Number theory: natural sqrt of 2nd degree polynomial above the naturals.

I am trying to find solutions to $y =\sqrt{ax^2+bx+c}, x \land y \in \mathbb{N}$. I've looked at Sqrt of polynomial, How to find integer X that gives integer y but the solution there assumes $a=1$, ...
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1answer
140 views

Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$?

Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$? Let, $2a-b=k^2, a-2b=p^2, a+b=q^2$. $k^2=p^2+q^2$ after ...
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130 views

Why is the sqrt of a matrix different to ^0.5

If H = [4 4; 4 4] Why is sqrt(H)=[2 2; 2 2] And, (H)^0.5=[1.4142 1.4142; 1.4142 1.4142]
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If a nine digit number is formed by the nine non zero digits with units digit $5$, prove that it must not be a perfect square. [duplicate]

If a nine digit number is formed by the nine non zero digits with units digit $5$, prove that it must not be a perfect square. Greetings, I was continuing on with my number theory, and a question ...
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53 views

How to prove that the difference of squares of two natural numbers can't be a perfect square if their sum is a perfect square? [duplicate]

I came across the following problem and wrote up the following proof a few years ago - what do you think? It is required to demonstrate the truth of Statement S1: "If $a^2 + b^2$ is a perfect square, ...
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$a, b$ are the integer part and decimal fraction of $\sqrt7$ find integer part of $\frac{a}{b}$

$a, b$ are the integer part and decimal fraction of $\sqrt7$ find integer part of $\frac{a}{b}$ using calculator : $\sqrt 7$ = 2.645 $\frac{a}{b} = \frac{2}{0.6} = \frac{20}{6} = 3.333$ integer part ...
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1answer
269 views

Any positive integer can be written as sum / difference of consecutive squares

How should one go about proving that $x \in \mathbb{N}$ can be written (with the right combination of signs) as $\pm 1^2 \pm 2^2 \pm \ldots \pm n^2$ for any $x : x, n \in \mathbb N$? I have tried for ...
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3answers
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Which primes $p$ satisfy $n^2 \equiv -1 \mod p$ for a perfect square $n^2$? [duplicate]

I am trying to solve a homework exercise in elementary number theory: Which primes $p$ satisfy $n^2 \equiv -1 \mod p$ for a perfect square $n^2$? After looking at the case $p=5$, I saw that $3^2 \...
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It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square

Prove that it is impossible for three consecutive squares to sum to another perfect square. I have tried for the three numbers $x-1$, $x$, and $x+1$.
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Is $100$ the only square number of the form $a^b+b^a$?

Conjecture $100$ is the only square number of the form $a^b+b^a$ for integers $b>a>1$. In other words, $(a,b)=(2,6)$ is the only solution. Can we prove/disprove this? Observations The ...
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1answer
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packing uniform cuboids into regular cube

I have a pile of uniform cuboids, of side a,b,c. I would like to make a regular cube. The sides are not in harmonic ratio ( cf. de Bruijn) but are in fact white sugar lumps. The minimum regular cube ...
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2answers
96 views

If $q$ is prime, can $\sigma(q^{k-1})$ and $\sigma(q^k)/2$ be both squares when $q \equiv 1 \pmod 4$ and $k \equiv 1 \pmod 4$?

This is related to this earlier MSE question. In particular, it appears that there is already a proof for the equivalence $$\sigma(q^{k-1}) \text{ is a square } \iff k = 1.$$ Let $\sigma(x)$ ...
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5answers
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Parametric solution of $ \alpha x^2 +\beta y^2 = \gamma w^2+\delta z^2$ quadratic Diophantine equation

I'm looking for a method to find the parametric solutions of a Diophantine equation of this kind $$ \alpha x^2 +\beta y^2 = \gamma w^2+\delta z^2 $$ $(\alpha,\beta,\gamma,\delta,x,y,w,z) \in \...
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1answer
45 views

Simple, algebraic issues.

I'm doing some geometry exercises. The first thing I did was inscribe a circle within a square (gave them the same diameter) and found the ratio between the areas to be Circle/Square=$\pi$/4. The ...
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1answer
86 views

Squares in different bases

In base 5, 121 is 25+10+1=36=6*6 In base 6 121 is 36+12+1=49=7*7 In base 7 121 is 49+14+1=64=8*8 In base 5 144 is 25+20+4=49=7*7 In base 6 144 is 36+24+4=64=8*8 Can someone please explain why ...
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0answers
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Given that a is an integer, find possible $a$'s so that $2^{2a+1}+2^a+1$ is a perfect square [duplicate]

I found this question in an old Olympiad from my country, but the solution I found only provides a brute force approach. The solution said $0$ and $4$, and when it reaches the point it tests $8$, ...
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1answer
36 views

Number of N-digit perfect squares and all its members [closed]

I've been thinking about this for a few days, and I was wondering if there are resources out there that describes this? here is what I have so far. $$ \text{Let}\; \mathbb{N} \;\text{be the set of ...
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1answer
88 views

Perfect Squares in Fibonacci Numbers [duplicate]

Let us define Fibonacci Numbers as: $F_1=F_2=1,F_n=F_{n-1}+F_{n-2}$ for $n>2$. $F_1$,$F_2$, and $F_{12}$ are perfect squares. Find the least integer $n>12$, if any, such that $F_{n}$ is a ...
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2answers
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How do I solve this question? I need help. [closed]

I can't understand how to solve this question. Please help me. $\left[3({x^{1/3} - x^{-1/3}})\right]^{1/3} = 2,\; \text{then}\; x^{1/3} + x^{-1/3} = ?$
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1answer
64 views

Pell's equation solution set [closed]

Can we guarantee that pell equation has infinitely many solutions in positive integers without finding a non trivial solution?
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61 views

Square Root of the Square of a Negative Number

We define Square Roots as $$\sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \\ -x, & \mbox{if }x < 0. \end{cases}$$ However, if we take the Square Root of the ...
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1answer
102 views

Diophantine $\frac{a^2 + b^2}{ab + 1} = \frac{c^2}{d^2} $

Consider the Diophantine equation with $a,b,c,d > 0$ : $$\frac{a^2 + b^2}{ab + 1} = \frac{c^2}{d^2} $$ For the case $d=1$ , this is a Classic ; we know that Diophantine $\frac{a^2 + b^2}{ab + 1}...
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3answers
696 views

How to work out sum of Square Number from given numbers?

Which of the following cannot be written as the sum of two distinct square numbers? A.106 B. 109 C. 112 D. 117 What would be the correct answer here and can someone explain in detail please.
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2answers
66 views

When can an even number be written as the difference between two squares? [duplicate]

When(What properties should a number have) can an even number be written as the difference between to perfect squares? Here's what I've tried: Let $n$ be that number and let $x^2$ and $(x+y)^2$ be ...
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4answers
82 views

When Does There Exist an Integer $n$ So $a_1 + n, a_2 + n, …,a_9 + n$ Are All Perfect Squares for $9$ Distinct Natural Numbers?

Let $a_1, a_2, ...,a_9$ be $9$ distinct positive integers. My question is, when(What properties should $a_i$ have) does there exist an integer $n$ so $a_1+n,a_2+n,...,a_9+n$ are all perfect squares? ...
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3answers
123 views

Square valued integer polynomial

For what integer values of $n$ is the expression $n^{6}+n^{4}+1$ a square? This is a square when $n=2$; when $n$ is odd, this expression is $3 \ mod(8)$ and so cannot be a square.
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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 ...
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2answers
78 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 $(1, 1)$...
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1answer
57 views

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, ...
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4answers
187 views

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 ...
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1answer
739 views

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.
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2answers
25 views

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 ...
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0answers
14 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 ...
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2answers
772 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:& ...
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1answer
167 views

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 ...
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0answers
80 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 ...
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1answer
63 views

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 ...
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1answer
73 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. $a$, $b$, ...
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1answer
45 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 ...
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1answer
236 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 ...
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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 ...
0
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3answers
41 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?
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1answer
332 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

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