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

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seemingly simple question

the largest number which cannot be expressed as the sum of any number of distinct squares. But it is divisible by the total number of its divisors, making it a refactorable number
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39 views

Is the square root of -1 really “i” [duplicate]

I know that the imaginary unit i is a number with the following property: i^2 = -1 But I often see people turn that into this ...
1
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2answers
21 views

A property regarding complete/perfect squares.

When a set of natural numbers is under consideration, if we add first consecutive 'n' odd natural numbers(i.e. from 1 ) we get a complete square whose root is 'n' itself. ...
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2answers
86 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
2
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0answers
61 views

Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...
4
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1answer
66 views

Elementary proof that $\sum_{k=1}^{\infty}{\frac{1}{m^{k^2}}}$ is irrational (for any integer $m > 1$)

I used similar technique as Fourier's proof of irrationality of $e$ https://en.wikipedia.org/wiki/Proof_that_e_is_irrational to show that this series is indeed an irrational number but I was wondering ...
3
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2answers
58 views

If $2xy$ divides $x^2+y^2-x$, prove that $x$ is a perfect square [duplicate]

This problem is from ( BMO Exam1991 ). I tried to solve but it was difficult. The problem is: If $ x^{2} + y^{2} - x $ is a multiple of $ 2xy $ where $x$ & $y$ are integers, prove that $x $ ...
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2answers
31 views

Simple question, what is meant by 'as $x \to \infty$ the number of squares $\leq x$ is $\sqrt{x} + O(1)$?

For $x \to \infty$: the number of squares $n^2 \leq x$ is $\sqrt{x} + O(1)$. From here (page 6). More specifically, do they mean that... I'm confused now. I'm really not sure what they mean ...
16
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2answers
240 views

What is this pattern found in the first occurrence of each $k \in \{0,1,2,3,4,5,6,7,8,9\}$ in the values of $f(n)=\sqrt{n}-\lfloor \sqrt{n} \rfloor$?

Learning how to generate the Mandelbrot set, I came across the definition of the "escape condition" which is the one that decides the color that is applied to each point of the plane where the ...
2
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1answer
44 views

Infinitely many correct solutions to equation? [duplicate]

I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below: $4!+1=5^2$ ...
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1answer
52 views

Square root of whole number number of solutions [duplicate]

Hi The GRE prep test is asking for the square root of a number.. for example $\sqrt{16}$. It says the answer is $4$. Couldn't the solution be both $4$ and $-4$?
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36 views

Bounded pythagorean triples

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
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0answers
69 views

Proving that $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is never a nontrivial integer square?

I'm trying to prove that if $a,b$ are integers, and $b^6-2a^3b^3-2b^3+a^6-2a^3+1$ is a square [integer], then $ab=0$. What general tools are available to attack such a problem?
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2answers
24 views

Simplifying some square numbers expressions

I need help to fix theese out. Thank you. $ \frac { \frac {1} {\sqrt {3} } - \sqrt {12} } { \sqrt {3} } $ $ \frac { \sqrt {x} } {\sqrt[3] {3} } + \frac {\sqrt[4] {x}} {\sqrt {x} } $ $ \sqrt {5} ...
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2answers
52 views

A basic square numbers equation

I couldn't solve this equation. Could you help me to solve this step by step. Thank you. $$\sqrt[4]{\frac{8^x}{8}}=\sqrt[3]{16}$$
2
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1answer
100 views

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one ...
2
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1answer
26 views

What is the relationship between number of perfect squares and number of perfect cubes?

If S is the "number" of perfect squares between 1 and a specific integer N and Q is the "number" of perfect cubes between 1 to N. How we can show the relationship between S and Q? Please explain with ...
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9answers
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Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$

The formula basically is: The sum of all integers before and including $n$, plus all the integers up to and including $n-1$. This will find $n^2$. $$ \sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2 $$
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1answer
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Have I just discovered an easy way to square numbers?

Choose any number, $x$: say, $x = 876$ (you can pick any $n$ digit number) Now, square the number -> $876 * 876 = 767376$ But now, If I ask you the square of $ x + 1$ --> $876 + 1 = 877$. You can't ...
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1answer
25 views

Powers of numbers containing negative numbers [duplicate]

I just came across this question and thought if i could ask help. How do you solve problems that have powers with a negative number? Ex. 2^(-2)
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1answer
28 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}$ is a rational square where $ \sigma(k) $ and $k$ both are square?

Is There some one who can show me if there are infinitely many $k$ for which $$\frac{\sigma(k)}{k}$$ is a rational square where $\sigma(k)$ and $k$ both are square ? Note :$\sigma(k)$ is sum ...
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2answers
59 views

This expression is always a perfect square [closed]

How to show that for $x,y\in \Bbb R$, the expression $xy+\left(\frac{x-y}{2} \right)^2$ is always a perfect square? For example $x=7, y=3$, $7\times 3+\left(\frac{7-3}{2} \right)^2=25=5^2$
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Digital Roots of Square Numbers

Can anyone offer a proof of the following: The digital root of a square number is always $1$, $4$, $7$ or $9$. (It is never $2$, $3$, $5$, $6$ or $8$.) Digital root : Add the digits of a number ...
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5answers
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Simplify Square Root Expression $\sqrt{125} - \sqrt{5}$

$\sqrt{125}-\sqrt5$ simplify it. I thought it would be $\sqrt {5\cdot5\cdot5}-\sqrt 5$ which would be the square root of 25 which is 5 but it is not. Can you show how to simplify this?
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6answers
343 views

How to solve “$4\sqrt5$ is the same as which square root?”?

What is the right method for solving a problem like this: ”$4\sqrt{5}$ is the same as which square root?" Possible answers are: $\sqrt{20}$ $\sqrt{10}$ $\sqrt{40}$ $\sqrt{80}$ I have been ...
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0answers
17 views

Prove that if $\ p^2 = a^2+2b^2 $ then $\ p = m^2+2n^2 $ (where a, b, m, n are integers, and p is prime) [duplicate]

Given that $\ p^2$ can be written in the form $\ p^2=a^2+2b^2 $ (where a & b are integers, and 'p' is a prime number), then prove that the prime number 'p' can also be written in the form $\ ...
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If the square of a number is even, then the number if even. Isn't that not true for 2?

I'll quickly go over my understanding of it: If a number $n^2$ is even, then $n$ is even. The contrapositive is that is that if $n$ is not even (odd), then $n^2$ must also be not be even (be odd). ...
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3answers
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How many of the numbers in $A=\{1!,2!,…,2015!\}$ are square numbers?

Problem How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers? My thoughts I have no idea where to begin. I see no immediate connection between a factorial and a possible square. ...
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0answers
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What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
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3answers
93 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...
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2answers
50 views

Square numbers in the form $1+4y$

I want to solve the equation $y+x=x^2$: $$ x^2-x-y=0 \\ x_{1;2}=\frac{1\pm \sqrt{1+4y}}{2} $$ However I want the solutions to be only natural numbers; the question then turns to find values of $y$ ...
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45 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
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63 views

Find all $n$ such that $n^2+3^n$ is a square number [closed]

Find all $n$ such that $n^2+3^n$ is a square number .
2
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2answers
66 views

Are there any primes for which $a^2 = pb^2 + 1$ does not exist?

The smallest solution to the above equation for various primes are: $(p=2)$ $3^2 = 2*2^2 +1$ $(p=3)$ $2^2 = 2*1^2 +1$ $(p=5)$ $9^2 = 5*4^2 +1$ $(p=7)$ $8^2 = 7*3^2 +1$ Is there at least one ...
12
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4answers
264 views

Prove or disprove that $8c+1$ is square number.

Let $a,b,c$ be positive integers, with $a-b$ prime, and $$3c^2=c(a+b)+ab.$$ Prove or disprove that $8c+1$ is square number.
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If we do not know a number's factors, what is the algorithm (if there is one) to write it as a difference of two squares?

For example, if we have a number like 29873412895, is there an algorithm that can find it as a difference of two squares? Or must you need the factors of the numbers? And what might be the algorithm? ...
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2answers
115 views

Is there an algorithm for writing an integer as a difference of squares?

For example, if we have $36$, is there an algorithm to determine that it may equal $10^2-8^2$? What if we blow up the number to something like $492709612098$? Can it be written as the difference of ...
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2answers
49 views

Please help me solve for $L-L^2$?

Unfortunately, I don't know any basic maths, and I need to solve the following equation for $L$ using the intercept of my graph: $$\begin{aligned} intercept & = \frac{2L}{\pi^2 + L(L - ...
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4answers
260 views

When is $8x^2-4$ a square number?

I asked an earlier question on when $32x+32$ is a square number (here) and I got a very clear answer. Now I am looking to solve for which $x$ the equation $8x^2-4$ results in a square number. When I ...
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3answers
174 views

When is $32x+32$ a square number?

I am trying to find out for what values for $x$ does the function $f(x)=32x+32$ return a square number? I found that this is the case for at least: $x \in ...
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2answers
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Proof that irrational coprime square root sums and products are always irrational?

I probably phrased it very bad. This is what I mean: $$\sqrt{x} + \sqrt{y} \neq R$$ x and y being non-square coprime natural numbers. And: $$\sqrt{xy} \neq R$$ x, y, AND R being coprime. Let's try to ...
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7answers
55 views

Square rooting to make quadratic function

If I have $x^4 - 34x^2 + 225 = 0$, is it not possible to to square root both sides of the equation so that I now have $x^2 - 34x + 15$? If this is true, then how would I go about solving the equation ...
2
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1answer
43 views

Square of One more than the Larger Part

This isn't really much of an algebra problem but different people are giving me different interpretations of " Square of One more than the Larger Part ". I've been given " Separate 17 into two parts. ...
25
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3answers
407 views

Can a number be equal to the sum of the squares of its prime divisors?

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ then define $$f(n):=p_1^2+\cdots+p_k^2$$ So, $f(n)$ is the sum of the squares of the prime divisors of $n$. For which natural numbers $n\ge 2$ do we have ...
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Very tentative proof of Beal's Conjecture?

I'm a high school student, so please point out my mistakes nicely and in layman's terms :) Thanks! Ok. Beal's Conjecture: If $$a^x+b^y=c^z$$ where $a$, $b$, $c$, $x$, $y$, $z$ are whole numbers; $x, ...
5
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3answers
70 views

Is it possible to have $a^2 + b^2 = c^2 + 1$ for $a$, $b$, $c$ being coprime integers?

As stated above. I'm working on a possible proof. It appears that $$(b+1)(b-1)=(c+a)(c-a)$$ That's where I'm stuck. Any help please? A clear, simple proof desired, thanks!
2
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4answers
120 views

I call them squares. They called them arrays. What do they mean?

So I was in C++, and we had third graders come today to play our programs. Whilst the others just drilled them with problems, my game was subtract a square. It was fun watching them discover that ...
2
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2answers
83 views

Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
6
votes
2answers
105 views

When is the sum of divisors a perfect square?

For $n=3$, $\sigma(n)=4$, a perfect square. Calculating further was not yielding positive results. I was wondering is there a way to find all such an $n$, like some algorithm? We know that if ...
0
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2answers
33 views

Continued addition and under rooting of 12

$\sqrt{(12 + \sqrt{12......})}$ and so on.... How do I find its answer? This is a question in our class VII mats book. P.S. - Answer is 4