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

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Square root Question from GRE [on hold]

A question in GRE states: What is the smallest number which when subtracted from 1.00060219 gives a perfect number? Any easy method & time saving please Update: This was what presented in a ...
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20 views

Estimate error on square root simplification

I have following term: $\sqrt{(\gamma+2)^2+4\gamma}$. I know that I could be able to simplify it to: $\sqrt{(\gamma+2)^2+4\gamma}$ $\approx$ $(\gamma + 2) + 2\sqrt{\gamma} + \epsilon$ and that this ...
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5answers
40 views

Intuition to why average of the square of a positive integer and the integer itself is the sum of all numbers from 1 to the integer?

The sum of all numbers from 1 to n, i.e. $\sum_{i=1}^n i = \frac{n(n+1)}{2} = \frac{n^2 + n}{2}$ This happens to be show that the average of a number and its square equals the sum of all numbers ...
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3answers
54 views

Find 3rd and 4th co-ordinates for a square given co-ordinates of two points?

To construct a square we need 4 points . In my problem 2 points are given we can find 3rd and 4th point . e.g. A (1,2) B(3,5) what should be the co-ordinate of 3rd (C) and 4th (D) points . Please ...
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1answer
29 views

Possible length of isosceles triangle side.

The perimeter of a right triangle $RST$ is equal to the perimeter of isoceles triangle $xyz$ The lengths of the legs of the right triangle are 6 and 8. If the length of each side of the isoceles ...
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1answer
14 views

Is there any polygon exists such that sum of length of square of two adjacent sides is equal to another side/diagonal?

In Right angle triangle we have $ a^2 + b^2 = c^2$ where $a^2 = (x_1-x_2)^2 + (y_1-y_2)^2 ,$ $b^2 = (x_3-x_2)^2 + (y_3-y_2)^2 $and $c^2 = (x_1-x_3)^2 + (y_1-y_3)^2$ And in Square we have $ a^2 + ...
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2answers
758 views

Discrete math. Finding a perfect square.

The problem is: Find all natural numbers $n$ for which $2^n + 1$ is a perfect square? I am having a bit of trouble finding a generic way of finding these numbers. Of course the first obvious solution ...
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4answers
1k views

There exist no integers for which $x^2-4y=2$

I am working on a new exercise in my textbook: $$\text{Prove that: (P): }\;\nexists \;x,y \in \mathbb{Z}, x^2-4\cdot y = 2 $$ I am stuck and I would really like to see a correct proof so I can move ...
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128 views

finding the difference of perfect squares

Find the difference between the smallest perfect square larger than one million and the largest perfect square smaller than one million. I did not want to use a calculator for this question. I ...
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1answer
161 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
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2answers
21 views

Growth Rate of Gaps Between Consecutive Perfect Squares

What is the best mathematical expression for this? For any pair of consecutive perfect squares, the quantity of integers in the interval between the two squares is equal to twice the square ...
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0answers
26 views

Reducing a sum of four squares to a sum with root sum equal to $1$

It is well-known that every odd natural number can be written as the sum of four squares. Perhaps less well-known is the fact that every odd natural number can be written as the sum of the squares of ...
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0answers
25 views

What does Heron's Algorithm have to do with the construction of logarithmic tables

i need a little help answering this question, what does Heron's Algorithm have to do with the construction of logarithmic tables. I know that Heron's algorithm is used for finding square roots, but ...
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1answer
34 views

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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2answers
64 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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16 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
84 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
30 views

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
101 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
43 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
64 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
78 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
360 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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40 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
56 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
47 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
28 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
52 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
155 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
108 views

Killer Tree! (one of my old questions)

There is a problem that killed me! but I couldn't solve it: We have a tree graph witch its structure is what is on image. Proof that there is no reduplicative numbers in each line.
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1answer
24 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
134 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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158 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
104 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
173 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
86 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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53 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
79 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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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
199 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
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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
37 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
62 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
44 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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$a^3+3a^2+a$ 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
46 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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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 ...