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

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Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function.

The following is an exercise in my textbook on complex analysis: Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function. ...
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2answers
69 views

How to proove that $\log(n)$ < $\sqrt{n}$? [on hold]

How to prove that $\log(n)$ < $\sqrt{n}$ I understand that O($\log(n)$) should work faster than O($\sqrt{n}$) but I can't understand how?
10
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4answers
145 views

Prove that $2^n+3^n $ is never a perfect square

My attempt : If $n$ is odd, then the square must be 2 (mod 3), which is not possible. Hence $n =2m$ $2^{2m}+3^{2m}=(2^m+a)^2$ $a^2+2^{m+1}a=3^{2m}$ $a (a+2^{m+1})=3^{2m} $ By fundamental ...
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2answers
33 views

Square Number Problem

If one is given a set of digits, such as 3352, is there a simple way of finding the square numbers closest to 0 that would begin with the given digits?
2
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3answers
52 views

Do you write plus/minus if a variable squares equals the square root of a number?

For example, if I have $x^2 = \sqrt{49}$. I know that $7$ is the number, but as my final answer, do I write that $x = +\sqrt{7}$ and $-\sqrt{7}$ or just $x = \sqrt{7}$?
1
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1answer
79 views

Sum of two consecutive squares equal square

$N^2 + (N+1)^2 = K^2$, find all solutions for $N<200$ I have done some factoring and also realized that $ K=[n\sqrt{2}]+1$ in eventual solutions, where $[x]$ denotes the greatest integer less ...
1
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1answer
50 views

Is it correct to say that Square Root of 0.9 is 0.9 itself

When I calculate Square Root of 0.9, it comes around 0.9486832980505138. Though I have heard people occasionally saying that the square root of 0.9 is 0.9 itself. Would it be correct to make a ...
0
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2answers
28 views

Pattern in digits of sums of consecutive squares

I am interested in patterns in square numbers as well as the reasons behind them and I can't seem to figure out (also how to prove) why do the sums of two consecutive squares only end in digits 1, 3 ...
1
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1answer
56 views

hailstone sequence of perfect squares (collatz conjecture)

The Collatz conjecture states: Take any positive integer $n$. If $n$ is even, divide it by $2$ to get $n/2$. If $n$ is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process ...
2
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0answers
67 views

$m^2+n^2$ and $m^2-n^2$ cannot both be squares [duplicate]

I need to show that there aren't any $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both squares. First of all, assume without loss of generality that $m$ and $n$ are co-prime, since otherwise we ...
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2answers
48 views

Does a square root come out plus/minus even if there is a negative sign outside?

For example: $-\sqrt{100x^{20}y^{10}}$. Would that give $\pm10x^{10}y^5$ or just $-10x^{10}y^5$?
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0answers
59 views

Sum of three consecutive cubes equals a perfect square

I have found this problem in an old German textbook: Find all sets of three consecutive integers such that the sum of their cubes is a perfect square. We can write $$S = (x-1)^3 + x^3 + (x+1)^3 = ...
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3answers
77 views

Calculating square roots without a calculator [closed]

Can anyone help me calculate $\large{\sqrt{\frac{4}{11}}=\sqrt{0.\overline{36}}}$ using the digit by digit method?
4
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1answer
64 views

Prove that for $k>1$ $a_n$ is a perfect square

I'm having problems with this exercise. Let $k > 1$ be an integer. We define $(a_n)_{n \in \Bbb N_0}$ as: $$a_0 = 1$$ $$a_1 = 1$$ $$a_{n+2} = (k^2-2)a_{n+1}-a_n-2(k-2)$$ Prove that $\forall n \in ...
3
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4answers
67 views

$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers

$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ My attempt $\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$ $(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$ $x+938^2 + ...
2
votes
3answers
73 views

Finding a number $n$ and $k$ such that $nx+k$ will be a perfect square for any two given $x$.

Given two positive integers $x_1,x_2$, is it always possible to find positive integers $n$ and $k$ such that the expression $nx_i+k$ becomes a perfect square for each $i$ ?
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0answers
22 views

What triples of square-free integers $(r,s,t)$ admit integer solutions $(x,y,z)$ where $rx^2,sy^2,tz^2$ are consecutive integers?

In this post on the consecutive integers $b^2,2a^2,3c^2$, I asked whether the trivial solution $a=b=c=1$ was the only one. At this time, that question appears to have been answered in the affirmative ...
2
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2answers
36 views

Counting Squares In a Range

I've been working on a series of programming challenges to work on my math skills, and I came across a solution that I don't know how to explain. The problem: "For ...
2
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1answer
50 views

variant of Lagrange's four square theorem using a restricted set of squares

The well-known four square theorem states that any positive integer is the sum of at most four squares. Suppose that, instead of allowing all squares, I only consider the following set of squares: $$ ...
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1answer
59 views

Square Root of $5$ mod $10^{9}+7$ [closed]

$My$ $Current$ $Knowledge:$ We can find it if 5 is a $Quadratic$ $residue$ modulo p and where p is prime and we can check it using $Euler$ $criterion$. I cannot able to find the root(5)mod 1000000007. ...
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0answers
19 views

Finding large integer squares - how to decide modulus to calculate.

Say I want to decide if an integer is a square. My integer is rather big so I can't keep it in memory altogether, but I know that it is a product $$f = \prod_{i=1}^nf_i, f_i\in \mathbb{N}$$ Then I ...
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0answers
22 views

How to prove that $gxyz$ and $g(y-x)$ are perfect squares? [duplicate]

Let $x,y,z$ be positive integers such that $1/x-1/y=1/z$. Let $g=\gcd(x,y,z)$. Prove that $gxyz$ and $g(y-x)$ are perfect squares.
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1answer
93 views

How many perfect squares exist? [closed]

Consider a set of $1985$ positive integers not necessarily distinct. Every number in set can be written in the form $p_1^{{\alpha _1}}p_2^{{\alpha _2}} \cdots p_9^{{\alpha _9}}$ where ...
0
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1answer
73 views

Primes Between Squares of Primes

Is this problem still open? I know that Henri Brocard conjectured that there are at least four primes in the interval between each pair of consecutive squares of primes from nine onward. ...
1
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1answer
55 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n
1
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1answer
61 views

Why this book says that $ 2^{1/2} = ±\sqrt{2} $?

Shouldn't it be: $ 2^{1/2} = \sqrt{2} $ ? I know the problem is that there they are working with complex numbers, but I still don't understand. The book is in the link, page 113, when they move ...
1
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0answers
28 views

Floor Function and Euler-Mascheroni Constant

I need to know how to express the following function, $\lfloor{\sqrt{c^2-707}}\rfloor^2=y$ $(s.t.$ $c $ and $y$ are in $N)$, analytically. I think ...
0
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1answer
128 views

Different proofs for $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ [closed]

Different proofs that show $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ cannot be the square of an integer, where n is a natural number.
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1answer
28 views

Square Root of Rational Number $\frac{A}{B}$

Here's the question: Let $x=\frac{A}{B}$ be a positive rational number in lowers terms (i.e., $A, B\in\mathbb{N}$ and $hcf(A,B)=1$). Prove that $\sqrt{x}$ is rational if and only if $A$ and $B$ are ...
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4answers
95 views

Prove that $x^2+1$ cannot be a perfect square for any positive integer x?

I started this problem by trying proof by contradiction. I first noted that the problem stated that $x$ had to be a positive integer, and thus $x=0$ could not be a solution. I then assumed that ...
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2answers
76 views

Find all positive integers n such that $2^2 + 2^5+ 2^n$ is a perfect square. [closed]

Find all positive integers n such that $2^2 + 2^5 + 2^n$ is a perfect square. Explain your answer.
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462 views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\frac{1}{19^2}\Big)} = ...
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1answer
42 views

What is the relation between the square root of the sum of squares and the sum of the absolute values?

I want to prove that $\sqrt{\sum a_{i}^{2}} \geq \sum \left | a_{i} \right |$, is it possible ?
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4answers
70 views

When will the sequence $k \mapsto A + Bk + k^2$ yield a perfect square?

Consider the following sequence: $$a(k) = A + Bk + k^2 ,$$ where $A$ and $B$ are both integers, and $A < B$ ($k$ is of course an integer variable, B is even). Problem: For which $k^*$ is ...
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1answer
46 views

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares.

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares. First few numbers are ...
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2answers
38 views

How can I make a square from these terms? [closed]

Can anyone help me to make a single square from these terms below: $$\frac{(x-a)^2}{A} + \frac{(x-b)^2}{B} = \frac{1}{AB}\left((A+B)x^2 - 2(aB + bA)x + a^2B + b^2A\right)$$ Thanks.
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0answers
23 views

Distribution of squared multivariate normal random variable

Let $W\sim MVN(\mu, \Sigma)$, here $W$ and $\mu$ are $k\times 1$ vector and $\Sigma$ is $k\times k$ symmetric matrix. And the diagonal elements of $\Sigma$ are equal to one. For this multivariate ...
0
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1answer
19 views

Number of N-digit Perfect Squares

I was working on a programming problem to find all 10-digit perfect squares when I started wondering if I could figure out how many perfects squares have exactly N-digits. I believe that I am close to ...
3
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5answers
112 views

$(x+1)^2 + (y+1)^2 + xy(x+y+3)=2$

I've came across this problem some hours ago and, although it looks (and possibly is) just some algebra calculus, I can't get on the right track. Find $x$, $y$ integers such that $$ (x+1)^2 + ...
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3answers
1k views

Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number ...
3
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3answers
92 views

Write $x_n=22..244…45$ as sum of $2$ squares

I've recently came across this problem and, although I've spent time looking for a solution, I don't have any interesting ideas. Let the numbers $$x_1=25$$ $$x_2=2245$$ $$x_3=222445$$ and ...
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1answer
37 views

Generalize finding perfect squares by adding odd numbers

I was doing a iterator-based Sieve of Eratosthenes (in Swift). I was using the variant where the detector for prime X wouldn't be inserted until I counted up to X^2. Instead of multiplying each ...
1
vote
1answer
75 views

What is the proof for $\sqrt{-a}\times\sqrt{-b}\neq\sqrt{ab},\text{ where }a,b\in \mathbb{R}$

Having just learned about $i$ in my 10$^{th}$ grade classroom I'm interested in the proofs underlying the rules for algebraic manipulations with imaginary numbers; such an understanding will create a ...
2
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2answers
23 views

infinitely many perfect squares in $\{2n^2+kn+l\colon n\in\mathbb{N}\}$

is there a pair $(k,l)$ of natural numbers such that the set $\{2n^2+kn+l\colon n\in\mathbb{N}\}$ contains infinitely many perfect squares?
2
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3answers
399 views

How do you find the squares mod 23?

1.) Compute the squares modulo 23 as efficiently as possible. 2.) Show that $y^2 = 23x^2 + 7$ has no integer solutions. This is a two part problem on my review for number theory and I am a bit lost. ...
0
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1answer
37 views

Proving that $z^4-6z^2+4z-3 = y^2$ has only one integer solution

I'm trying to prove the following result. Conjecture. If $z$ is an integer, and $z^4-6z^2+4z-3$ is a square, then $z=3$. A quick check modulo $9$ shows that $z=9w+3$ for some integer $w$. So for ...
2
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2answers
34 views

Square root fraction confusion

I was doing math for school and got to something that really confused me. With having the rule $\frac{2}{4} = \frac{4}{8}$ (or some simular fraction equation) in mind, I got to the following confusing ...
3
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3answers
42 views

How many tokens would person A have under these conditions?

Persons A and B each have a positive integer number of tokens, and the number of tokens B has is a square number less than 100. B says to A, "If you give me all of your tokens, my total number of ...
5
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2answers
83 views

If $\frac1x-\frac1y=\frac1z$, $d=\gcd(x,y,z)$ then $dxyz$ and $d(y-x)$ are squares

Let $x, y, z$ be three non negative integer such that $\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{z}$. Denote by $d$ the greatest common divisor of $x, y, z$. Prove that $dxyz$ and $d(y-x)$ are ...
1
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0answers
23 views

Recall that an integer is said to be square-free if it is not divisible by the square of any prime. Prove that for any positive integer $n$… [duplicate]

Recall that an integer is said to be square-free if it is not divisible by the square of any prime. Prove that for any positive integer $n$, there exist $n$ consecutive nonsquare-free positive ...