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

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Half prime numbers?

I am wondering if there is a term for a number which is only divisible by its square root, one and itself? For examle $25$ can be divided by $1, 5$ and $25$. And $169$ with $1, 13$ and $169$. I am ...
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40 views

Perfect squares, cubes and fifth powers $ \leq 10^8 $

I've the following question: Find number of numbers $ \leq 10^8 $ which are neither perfect squares, nor perfect cubes nor perfect fifth powers. What I currently have is: Number of perfect ...
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0answers
14 views

Simple square-root calculation parsing /order of operations

I'm started highschool, and this came up today. Not really cared about the anwser, but could someone kindly recall to me, that what is the right way to "parse" this calculation: $$ ...
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3answers
47 views

How to find the square root of an imperfect square number, using shortcut method?

Suppose of the number sqrt(156934). Any help is appreciated ! Thanks
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2answers
23 views

Unit conversion not coming out right.

I'm trying to convert 20.7x10^(-4)m^2 to inches^2. And after that's done use it to figure the price at 3.25/lb^2. The answer is supposed to be $10.43. I have the conversion factor 1m = 39.37in. I've ...
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0answers
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Leibniz Binary Representation of Squares

Leibniz claims to have found patterns in the square numbers and their binary representations. I cannot see any patterns at all. Here are the first ten squares and their binary representations, can ...
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2answers
93 views

Calculate more digits of square root of 2?

How would I calculate the next digit of the decimal representation of square root of two? 1.4142135... Say I want to calculate the digit after 5... how would I do that? I am not allowed to use a ...
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2answers
40 views

The probability of $n$ being a square, given the units-digit in its decimal representation

Given a natural number $n\in[1,N]$, the probability of $n$ being a perfect square is $\frac{1}{\sqrt{N}}$. What would be the probability, if we knew the units-digit in the decimal representation of ...
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1answer
57 views

Is sum of square of primes a square of prime?

I would like to know if it has been proved that : There are no $a$, $b$ and $c$, all prime numbers, such that $a^2 + b^2 = c^2$ There are no $a$, $b$, $c$ and $d$, all prime numbers, such that $a^2 ...
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3answers
69 views

Simplifying radical expressions such as $\sqrt{80}$

I am having trouble simplifying a radical expression, such as say...$\sqrt{80}$. What I do is firstly, I do 80/2, then 80/3, then 80/4, then 80/5...etc until I find the largest number that can be ...
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2answers
280 views

A number is a perfect square if and only if it has odd number of positive divisors

I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it. For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ ...
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0answers
37 views

Archimedes' Apprxomation of Square Roots

Supposing a square root $\sqrt{X}$, let $x$ be the approximation of $\sqrt{X}$, then we get these 2 formulas to estimate $\sqrt{X}$: $x_{n+1}=\frac{x_n+\frac{X}{x_n}}{2}$ and ...
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3answers
54 views

(a+b)^1/2 another question is Square root (-4)^2=?

(a+b)^1/2 and Square root (-4)^2=? I'm new to learning algebra. I know what (a+b)^2 is. But then I thought what happens with ^1/2 or ^1/4. Can someone explain me? Also I have 2 questions in my book. ...
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3answers
46 views

How to square a number in binary system?

If I want to find the square of 111 (written in binary) what do I do? I'm confused and keep on hearing different answers.
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1answer
27 views

Integer as a product of a square and a square free integer

My question actually relates to an example given on p. 28 of Julian Havil's "Gamma". Discussing a proof of the infinity of primes due to Erdos, Havil writes: [Erdos] uses a counting technique ... ...
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1answer
51 views

a square in a finite field of odd order

GF(q) is a finite field of order q, where q is odd. Prove that $a\in GF(q), a\neq0$ has a root in $GF(q)$ iff $a^{(q-1)/2}=1$. I tried to prove it this way: Suppose a has a root in ...
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1answer
69 views

Square root of $\frac{2}{2^x}$; how do I find $x$?

I have this: $$\sqrt{\frac{2}{2^x}} = 9.313225746154785 \times 10^{−10}$$ (sqrt(2/(2^x))) How should I find $x$? I know it's 61 for this case, but I'd like to know how to solve it for when I don't. ...
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3answers
145 views

Filling a 40 x 40 grid with 3x3 squares

I'm supposed to find out the minimum number of 3x3 squares that will completely fill up this 40x40 grid where overlapping squares is acceptable. Each 3x3 square also has to coincide with the grid ...
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1answer
20 views

Maximum length of a representation of a number as an alternating sum of squares

Define a function $$\mathscr R: \mathbb N \to \mathbb N, \ \ \mathscr R(n) = \lceil \sqrt{n} \rceil ^2 -n.$$ IE. the distance of $n$ to the next prefect square. Sequence A068527 on OEIS. If $\mathscr ...
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3answers
133 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
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0answers
142 views

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > ...
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1answer
99 views

True or False: For $n>1$, $n!$ can never be a perfect square.

I am trying to solve the following: True or False: For $n>1$, $n!$ can never be a perfect square. I am thinking on the following lines: Any perfect square $N$ is of the form ...
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4answers
115 views

Find all the ways to express 225 as a sum of consecutive odd integers

Use your results to find the squares that can be added to 225 to produce another square. I started off by taking the 9 divides 225 with quotient 25. (25-8) + (25-6) + (25-4) + (25-2) + 25 + (25+2) + ...
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3answers
84 views

How many positive integers $n$ are there such that $2n$ and $3n$ both perfect squares?

How many positive integers $n$ are there, such that both $2n$ and $3n$ are perfect squares? I tried to use modular arithmetic, but I'm stuck.
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5answers
52 views

How to prove that $x^2≡2(\bmod 3)$ is not a complete square

Let $m$ be the product of first n primes (n > 1) , in the following expression : $$m=2⋅3…p_n$$ I want to prove that $(m-1)$ is not a complete square. I found two ways that might prove this . My ...
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2answers
78 views

Hard elementary-number-theory question on solve all $n$s that make $2^6+2^{10}+2^n$ a square numbe

I want to know all the nonnegative integer $n$ that makes $2^6+2^{10}+2^n$ be some other integer's square. I have tried it numerically for a range from $0$ to $1000$, and only $0,9,11,12,15$ returns ...
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2answers
86 views

A mathematical phenomenon regarding perfect squares…

I was working on identifying perfect squares for one of my programs regarding Pythagorean triplet. And I found that for every perfect square if we add its digits recursively until we get a single ...
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3answers
198 views

Perfect Square relationship with no solutions

I would like to show that for positive integers $a>b,c$ all greater than 1 such that $c\nmid a$, there are no solutions to the following equation: $$a^2+1=b^2(c^2+1)$$ As was pointed out in the ...
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1answer
37 views

Difference of two squares relationship

When do we have for $b|c$ that the following relationship holds where $a\neq c$? (all the variables are integers) $$b^2-1=a^2-c^2$$
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1answer
35 views

Is 2(2k-1) is a perfect square for positive integer k?

For positive integer $k$, let $M = 2(2k-1)$, which of the following must be true? (a) $M$ is not a perfect square for any $k$. (b) There are infinitely many $k$ such that $M$ is a perfect square. ...
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3answers
58 views

How to find the Square Root of a Polynomial

$4x^4 + 4x^3 - 11x^2 -6x + 9$ How do you find the square root of this polynomial? I really don't understand. Please provide an easy-to-understand explanation. Thanks.
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1answer
42 views

Help on a perfect square.

Consider a question, that xyxyxyxy cannot be a perfect square. How should i tackle this problem. All i use is it must be $0,1 ($mod $3,4)$ and then the math, are there any another beatiful ways ...
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6answers
105 views

Square root of $x = 5$ [closed]

Is it true that $\sqrt{x} = 5$ is unsolvable? Why? $\sqrt{x} = 5$ $(\sqrt{x})^2 = 5^2$ $x = 25$ But $\sqrt{25} = \pm 5$.
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2answers
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$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
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1answer
42 views

Coefficients of the expansion of $(x+a)^2$ makes a perfect square?!

I have no idea how this thought popped into my mind, but I noticed that the coefficients of $(x+1)^2$, when expanded, makes a perfect square. No, I am not talking about adding them (although that ...
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2answers
57 views

How to solve this kind of equation?

I have an equation (in my homework) of the form $a=\sqrt{x^2 + b^2} + \sqrt{x^2 + c^2}$ which I would like to solve for $x$. I am not sure how best to proceed. My thought is to square both sides ...
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0answers
44 views

Test Perfect Square of any Function

consider i have equation like this: f(x) = 5X^2 + 4 my question is, how can i check f(X) whether is a perfect square or not. But, suppose we can't squaring X because of computer data type ...
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2answers
285 views

The least perfect square, which is divisible by each of 21,36 and 66 is (options)

(a) 213444 (b) 214344 (c) 214434 (d) 231444 Any short method to solve this question in 1 min?
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1answer
47 views

Squares that cannot be shown as sum of squares

How many $n \in \mathbb{N}$ are there so that there exists no such $M \in \mathbb{N}$ so that $n^2 =\sum_{i=0}^{M}{a_i^2}$ for distinct $a_i \in \mathbb{N}$? Source: http://mishabucko.wordpress.com
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2answers
97 views

Square numbers multiplied by non-square numbers [closed]

If you multiply a square number with a non square number, the result is never a square number. Here, a square number is product of an integer with itself. Do you agree with this statement? If ...
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2answers
49 views

Determine real number exists for relation with square roots

We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$. I am to find whether a real number exists for this relation, and the real number that satisfies. I start by squaring both sides, which yields: $$x - 2 = 4x ...
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2answers
26 views

What is his algorithm of taking square of a 5 digit number?

I was watching this TED talk: Mathemagics and the performer, as his final trick, attempts to square a 5 digit number in a fairly short amount of time by as well thinking out loud (starting at 11:00, ...
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1answer
65 views

Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
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2answers
59 views

Is $(x^2+y^2+z^2)$ always a perfect square when $x=n-1, y=n, z=n(n-1)$?

This question is from a non-mathematician (a programmer...) so please excuse any poor terminology. I was writing some test data for some (X, Y, Z) coordinate to magnitude conversions. As you know, ...
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1answer
71 views

Find the Perfect Square

I came upon the following question in a recent district math test, and I have no clue how to solve it, besides using a calculator and doing some serious multiplication, but no calculators were ...
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4answers
360 views

Prove a square can't be written $5x+ 3$, for all integers $x$.

Homework question, should I use induction?.. Help please
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2answers
50 views

Number theory question - squares

Suppose I have $n,m,t$ positive integers such as $nm$ is a square and $mt$ is a square, how do I prove that $nt$ is also a square? I have said: $nm=k^2$, $mt=f^2$ so $nt=(kf)^2/m^2=(kf/m)^2$. I need ...
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1answer
36 views

Prove that there are no such odd numbers $x,y,z$ such that satisfy both $xy+1=a^2$, $yz+1=b^2$ and $xz+1=c^2$.

Prove that there are no such odd numbers $x,y,z$ such that satisfy both $xy+1=a^2$, $yz+1=b^2$ and $xz+1=c^2$. And, of course, $x,y,z,a,b,c\in\mathbb Z$. I've proven it myself but I want to see some ...
5
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1answer
169 views

Finding integer cubes that are $2$ greater than a square, $x^3 = y^2 + 2$ [duplicate]

I was given an example of a cube that is $2$ greater than a square number. The pair: $27$ and $25$. What's the best way to find further pairs ?
4
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1answer
91 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...