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Questions tagged [square-numbers]

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

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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
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 $(1, 1)$...
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1answer
48 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
177 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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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 $b$?...
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1answer
72 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
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 ...
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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 ...
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2answers
92 views

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 : $\sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$
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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:& ...
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1answer
49 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
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 ...
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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 ...
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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. $a$, $b$, ...
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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 ...
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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 ...
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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 ...
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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?
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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
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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 $b^2-a^2$. ...
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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 ...
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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?
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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 $d \cdot k=2n^2 \implies d=\dfrac {2n^2}k$ ...
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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!
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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: $m=1, n=0$ $m=1, n=3$
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2answers
137 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!\...
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2answers
33 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 ...
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0answers
26 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(...
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2answers
45 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 ...
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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($\theta$) $$\int(\sqrt{2sec(\theta)-\sqrt{4sec(\theta)^2-4}}) 2sec\theta d\theta$$ $$\...
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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 ...
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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 \...
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1answer
41 views

Finding solutions in $\mathbb{Z}_{+}$ [closed]

Find all the triplets of positive integers $(a, b, c)$ such that: $a^2+b+3=(b^2-c^2)^2$
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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.
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1answer
47 views

finding perfect squares solutions for the following case

I was working on a number theory problem and create a equation. I tried research on this, but tbh I don't even know what should I google for... Here's my cases. $$n = \sqrt{N * \frac{1+\sqrt{4k^2+1}}...
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1answer
36 views

Product of pairwise sum is perfect square

For which $n$ can we divide $1,2,\ldots,2n$ into $n$ pairs so that the product of the sum of the $n$ pairs is a perfect square? If $n$ is even, this is possible: match the first number with the last, ...
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1answer
32 views

Number multiplied by itself does not give a square number

The answer to this is probably very simple but while working on a question I was surprised to discover than a number multiplied by itself does not give the same answer as the same number squared (in ...
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2answers
118 views

$n$-th term of the series 1 27 125 1000

What will be the nth term of the series 1 27 125 1000 for $n = 1$, it is 1 for $n = 2$, it is 27 for $n = 3$, it is 125 for $n = 4$, it is 1000
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2answers
52 views

nth term of the series 1, 16, 24, 1024

What will be the formula for finding nth term of the series for eq for n = 1 it will be 1 for n = 2 it will be 16 for n = 3 it will be 100 for n = 4 it will be 1024 And am i doing it the ...
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4answers
92 views

Finding the mistake of my fake proof where $\pm-2=-2$ [closed]

I am wondering what has gone wrong in the following: $4=4 \iff \sqrt{4}=\sqrt{4}\iff\pm 2= \sqrt{4}\iff \sqrt{4}=2\;\text{and}\;\sqrt{4}=-2\iff \pm2=2$ (from substituting back in and obviously ...
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4answers
126 views

Find $n^2+58n$, such that it is a square number

I have the following problem. Find all numbers natural n, such that $n^2+58n$ is a square number. My first idea was, $n^2+58n=m^2$ $58n=(m-n)(m+n)$ such that $m-n$ or $m+n$ must be divisible by 29,...
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0answers
38 views

Algorithm to find square numbers with certain distance

Is there an efficient way to calculate for a given $c \in N$ two square numbers $x^2,y^2$ with $x^2-y^2=c$, without being able to factorize $c$? I was just thinking about RSA, where $n=pq$ is given. ...
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1answer
132 views

Determine all integers $i$ such that $(i-29)(i+29)$ is a square number

Determine all integers $i$ such that $$(i-29)(i+29)$$ is a square number. I’ve tried some substitutions but none of them worked... I think that the only solutions are $i=\pm 29$, but I still ...
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2answers
43 views

Does there exist an infinite geometric progression whose terms are all squares

I am aware of the fact that the squares don't contain an infinite arithmetic subsequence, but I was wondering if the squares contain an infinite geometric sequence. In other words, does there exist ...
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3answers
80 views

Why is $(5\sqrt{5p}-3\sqrt{5q})(5\sqrt{5p}+3\sqrt{5q}) \equiv 5(5p-3q)(5p+3q)$?

I was working on the difference of two squares, $125p^2-45q^2$ Writing my answer, $$(5\sqrt{5}p-3\sqrt{5}q)(5\sqrt{5}p+3\sqrt{5}q),$$ onto Pearson, I got a popup that said my answer was equivalent ...
2
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1answer
55 views

Number theory: square numbers of a given form

I made a proof in an undergraduate number theory class which uses this assumption to make a crucial step in the proof. Could someone tell me if it is correct and maybe explain why or why not? If I ...
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0answers
25 views

A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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1answer
38 views

Find amount of square integers between two points

I was recently trying to solve a programming problem on hackerrank and the problem description was that given two points $a$ and $b$ I was supposed to find amount of square integers in given range. ...
3
votes
4answers
100 views

For which integer $n$ is $28 + 101 + 2^n$ a perfect square?

This question For which integer $n$ is $$28 + 101 + 2^n$$ a perfect square. Please also suggest an algorithm to solve similar problems. Thanks Btw, this question has been taken from an Aryabhatta ...
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2answers
170 views

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 ...