Questions tagged [square-numbers]

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

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4
votes
3answers
142 views

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 $x+1$.
29
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3answers
479 views
+50

Conjecture: 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 ...
0
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0answers
18 views

Korselt numbers; definitions and properties [closed]

If $N$ is an $\alpha$-Korselt number, is $N$ must always be a squarefree , or there is cases can be have a square prime factor.
-1
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0answers
14 views

Korselt numbers and sets [closed]

When $N$ is a $K_\alpha$-number($\alpha$-Korselt number), is $N$ must always be a squarefree composite number? or $N$ may be contain a square prime factor?
1
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0answers
7 views

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 ...
7
votes
2answers
90 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 $\sigma(x)$ ...
0
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5answers
105 views

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 \...
1
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1answer
42 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 ...
1
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1answer
35 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 ...
0
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0answers
38 views

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 $8$, ...
0
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1answer
33 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 ...
0
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1answer
44 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 ...
-3
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2answers
46 views

How do I solve this question? I need help. [closed]

I can't understand how to solve this question. Please help me. $\left[3({x^{1/3} - x^{-1/3}})\right]^{1/3} = 2,\; \text{then}\; x^{1/3} + x^{-1/3} = ?$
0
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1answer
49 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?
0
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2answers
39 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 ...
1
vote
1answer
71 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}...
0
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3answers
44 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.
0
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2answers
63 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 ...
3
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4answers
73 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? ...
0
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3answers
72 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.
1
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2answers
79 views

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 ...
1
vote
2answers
56 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 $(1, 1)$...
1
vote
1answer
49 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, ...
3
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4answers
180 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 ...
3
votes
1answer
78 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.
0
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2answers
18 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 ...
0
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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 ...
14
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2answers
751 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:& ...
1
vote
1answer
51 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 ...
1
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0answers
69 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 ...
2
votes
1answer
62 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 ...
0
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1answer
66 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. $a$, $b$, ...
0
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1answer
41 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 ...
1
vote
1answer
82 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 ...
11
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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
39 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?
0
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1answer
46 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
0
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1answer
38 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 $b^2-a^2$. ...
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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 ...
1
vote
1answer
64 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?
0
votes
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 $d \cdot k=2n^2 \implies d=\dfrac {2n^2}k$ ...
1
vote
1answer
50 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!
0
votes
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: $m=1, n=0$ $m=1, n=3$
4
votes
2answers
146 views

Does digit $6$ always lead to $\ 25921=161^2\ $?

Consider prime numbers with the property that the product of the factorials of the digits plus $1$ is a perfect square, for example the prime $$30241$$ leads to the square $$3!\cdot 0!\cdot 2!\cdot 4!\...
1
vote
2answers
37 views

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$. I cannot give a proof to this, although I have try on ...
0
votes
0answers
27 views

Totient summatory function and other function yields how many perfect squares?

How many perfect squares does the totient summatory function yield? $$ \Phi(n)=\sum_{k=1}^{n}\phi(k). $$ How many perfect squares does this function yield? $$ \Lambda(n)= \sum_{k=1}^{n} \phi(k)\phi(...
1
vote
2answers
47 views

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 ...
-4
votes
1answer
64 views

Square root of integral question [closed]

How can I solve it? $$\int\sqrt{x-\sqrt{x^2-4}} dx$$ This is indefinite integral. I solve like this: x=2sec($\theta$) $$\int(\sqrt{2sec(\theta)-\sqrt{4sec(\theta)^2-4}}) 2sec\theta d\theta$$ $$\...
2
votes
3answers
86 views

$f(n)+f(m) = q^2$ always has a solution

Prove or disprove: Let $f$ be a non-constant polynomial with nonnegative integer coefficients. Then there exist $m,n \in \mathbb{N}$ such that $f(n)+f(m)$ is a perfect square. I'm just posting this ...
0
votes
0answers
37 views

Unknown value in the modular numerator

I'm looking for a mathematical solution to find unknown value in the numerator as follows: IF: $$b_1 + b_2 = 151$$ $$n = 86167$$ $$ (231 * 336 + b_1)^2 \mod 86167 = 151^2 $$ $$ (25 * 336 + b_2)^2 \...