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

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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
28 views

Rationalising Top Heavy Surds [on hold]

What is $$\dfrac{12 - 5\sqrt{3}}{\sqrt{3}}$$ expressed in the form $$a + b\sqrt{3}$$ where $a$ and $b$ are integers? Give the correct answer and your method.
3
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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
55 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 ...
3
votes
2answers
242 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
33 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 ...
2
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3answers
48 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
45 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.
0
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1answer
26 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 ... ...
1
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1answer
48 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. ...
3
votes
3answers
137 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 ...
1
vote
1answer
19 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
111 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 ...
5
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0answers
136 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 > ...
2
votes
1answer
95 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
91 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) + ...
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
77 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
81 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 ...
3
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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
36 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
34 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
56 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
41 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
100 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
126 views

$a^3+3a^2+a$ 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
40 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
56 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
222 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, ...
0
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1answer
63 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
58 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
68 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 ...
3
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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
3
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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 ...
2
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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
166 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
89 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 ...
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2answers
39 views

magical isoceles triangle and 13/15 ratio

It seems eerily magical that $\dfrac {13}{15}$ corresponds within $99.926$ percent accuracy to the height of an isoceles triangle the height of isoceles $= \sqrt {.75} = 0.8660254...$ and $\dfrac ...
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6answers
512 views

If $n$ is a positive integer and is not a perfect square

If $n$ is a positive integer and is not a perfect square, how do you prove that $n^{1/2}$ is irrational?
2
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1answer
77 views

How can I prove that it is not a perfect square?

A $10$-digit number has one $1$, two $2$'s, three $3$'s and four $4$'s as its digits in some order. Prove that it can never be a perfect square.
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1answer
66 views

Prove that for all $a,b\in\mathbb Z$ of opposite parity there exists a number $c\in\mathbb Z$ such that $c+ab$, $c+a$ and $c+b$ are perfect squares.

Prove that for all $a,b\in\mathbb Z$ of opposite parity there exists a number $c\in\mathbb Z$ such that $c+ab$, $c+a$ and $c+b$ are perfect squares. So we could prove that $c+ab=k^2$, $c+a=l^2$ and ...
5
votes
2answers
89 views

squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
2
votes
2answers
90 views

Prove that no $n,m, 0<n<m$ exist such that $m^2 +mn+n^2$ is a square number

Prove or disprove the claim that there are integers $n,m, 0<n<m$ such that $m^2 +mn+n^2$ is a perfect square.