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

learn more… | top users | synonyms

3
votes
1answer
47 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
votes
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
71 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 $i$ ?
2
votes
0answers
21 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
votes
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
votes
1answer
46 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: $$ ...
-1
votes
1answer
58 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. ...
0
votes
0answers
18 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 ...
1
vote
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.
1
vote
1answer
91 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
votes
1answer
53 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
vote
1answer
51 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
vote
1answer
58 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
vote
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 ...
-2
votes
0answers
21 views

regarding calculation of square roots

While computing square root of a no., why do couple the digits from the left end?How does the doubling of nos.(in the divisor) help? Why do we concatenate a digit after each computation. Thanks,
0
votes
1answer
121 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.
0
votes
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 ...
4
votes
4answers
92 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 ...
-1
votes
2answers
71 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.
29
votes
0answers
409 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)} = ...
1
vote
1answer
41 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 ?
1
vote
4answers
69 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 ...
1
vote
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 ...
-1
votes
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.
1
vote
0answers
20 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
votes
1answer
18 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
votes
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 + ...
10
votes
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 ...
4
votes
3answers
89 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 ...
1
vote
1answer
36 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
67 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
votes
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
votes
3answers
372 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
votes
1answer
36 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
votes
2answers
31 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
votes
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
votes
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
vote
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 ...
0
votes
1answer
43 views

How to graph a Quadratic equation.

I have an equation that goes: $0.0001x^2 - 0.22x + 197$. I'm not asking for the answer, but instead, how can I graph it without dealing with these insanely tough numbers.
15
votes
1answer
120 views

For which $n$ can $\{1,2,…,n\}$ be rearranged so that the sum of each two adjacent terms is a perfect square? [duplicate]

For which numbers $n$ can the sequence $1$ to $n$ be rearranged such that each pair of consecutive terms adds up to a perfect square? Can this be done on the set of natural numbers as well? Integers? ...
1
vote
1answer
36 views

How to find two square roots whose difference is greater than one.

How do you find the greatest $n$ such that the difference of its square root from some other integer is greater than or equal to one? For example : $$2011^{1/2} - n^{1/2} \ge1$$ What should be the ...
1
vote
0answers
47 views

Is there a general method to determine the [non-]solutions to $ax^4+bx^2+c=y^2$?

I am looking for general methods to solve, in integers $x$ and $y$, the equation $$ ax^4+bx^2+c=y^2 $$ where $a,b,c$ are given integers, and by "solve" I mean: (i) show there are no [non-trivial] ...
-1
votes
1answer
54 views

What is the largest six-digit number with an odd number of positive factors?

What is the largest six-digit number with an odd number of positive factors? So I know the number must be a perfect square, but how do I know six-digit number perfect squares? I'm pretty sure there's ...
1
vote
1answer
30 views

If $\sigma _{1}(n)\mid \sigma _{2}(n)$, does $n$ has to be a perfect square?

Let's say $\sigma _{1}(n)\mid \sigma _{2}(n)$. Can we say, therefore $n$ has to be a perfect square? How to show that?
0
votes
1answer
39 views

How to write -1 as a square in a finite field of characteristic 2

If $\mathbb{F}_{q}$ is a finite field of characteristic $2$, where $q$ is a power of $2$ and $\beta$ is a generator of $\mathbb{F}^{*}$, then I know that $-1$ is a square in $\mathbb{F}$, but how do I ...
1
vote
0answers
73 views

How to tell if $\sqrt{e^x}$ is an integer?

Is there a fast way to determinate if $\sqrt{e^x}$ is an integer if the only thing you know is that $e^x$ is an integer I use $e$ as mathematical constant $e$, but you can change it if it helps to ...
4
votes
2answers
94 views

Find all square numbers $n$ such that $f(n)$ is a square number

Find all the square numbers $n$ such that $ f(n)=n^3+2n^2+2n+4$ is also a perfect square. I have tried but I don't know how to proceed after factoring $f(n)$ into $(n+2)(n^2+2)$. Please help me. ...
3
votes
1answer
57 views

Can you find it? or Can you prove it? [duplicate]

$x!+1=y^2$, I found 3 solutions. They are $(4,5),(5,11),(7,71)$. Is there a $4$th solution?If not can you prove it?
0
votes
2answers
17 views

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$… [duplicate]

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$,Prove that $a$ is the square of an integer if and only if $a_i$ is even for each $i$. -The ...
0
votes
1answer
39 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...