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

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1answer
11 views

Existence of a Repeating Divisor

I have $n$ integers $a_1, a_2, a_3, .., a_n$ let $X = a_1*a_2*a_3*...*a_n$. I want to know a single integer $F$ such that $F^2$ divides $X$. It is told that there will be atleast one such $X$ and ...
1
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0answers
47 views

No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
3
votes
1answer
72 views

What is the first square in the sequence $4729494n+1$?

Today I found a strange phenomenon that I want to ask about. If $$f(n)=4729494n+1,$$ is square, where $n$ is positive integers. Then I found $n=4729492$, because $$f(4729492)=4729493^2$$ In fact,...
3
votes
2answers
79 views

$n=a^2-b^2$ iff $n \not\equiv 2(\mathrm{mod\ }4)$

I have to show that $n=a^2-b^2$ iff $n\not\equiv 2$ (mod $4$). Where $a$, $b$ are integers. I already got the explicit $(a,b)$ if $n\not\equiv 2$ (mod $4$). However, I am stuck with the other ...
4
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1answer
92 views

Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square

$\textbf{Question.}$ Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square One thing which I observed is the following. Since $5 \equiv 1 \pmod{4}$, this ...
8
votes
3answers
174 views

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$.

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$. I tried taking all possible combinations $\bmod 3$ and $\bmod 4$ and it has a solution only ...
0
votes
1answer
74 views

Find the $m$ such $a_{n+1}=a^5_{n}+487$ [closed]

Let $\{a_{n}\}$ be a sequence of positive integers, and suppose $a_{0}=m$. Further, $\{a_{n}\}$ satisfies $$a_{n+1}=a^5_{n}+487.$$ Find $m$ so that this sequence consists of square numbers for as long ...
1
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3answers
62 views

How do I prove that $\sqrt{9+4k^2}$ holds integer value only for $k=0$ and $k=2$?

I've faced that sort of a problem while solving some other problem and it made me stuck for a while. It's vital to me to prove that for any other integer $k$ there can't be an integer output, i.e. a ...
1
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2answers
50 views

Consecutive Square Numbers [closed]

The difference between the squares of two consecutive numbers is $23$. What are the two numbers?
2
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0answers
35 views

Pattern involving squares, primes, and remainders

I ran across a really neat pattern, wholly by accident. In advance, my questions are: Has this been discovered before? If so, where can I learn more about it? Why does this pattern work? Now for ...
0
votes
1answer
19 views

Perimeter/footprint and square meters

I'mIf a room is 8x8 meter the perimeter will be 32m, and 64m2. If add 2m on two sides and subtract 2m on two sides the room will be 10x6m. The perimeter, and footprint, will be the same for both rooms ...
1
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0answers
26 views

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. ...
10
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4answers
153 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 ...
0
votes
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
votes
3answers
55 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
81 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 than....
1
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1answer
54 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
votes
2answers
32 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
58 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 ...
3
votes
0answers
70 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
49 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 $-10x^{10}y^5$?
7
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0answers
62 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 = (x-...
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votes
3answers
83 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
votes
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
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*\sqrt{x+938^2}*\sqrt{...
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 $i$ ?
2
votes
0answers
23 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
38 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
51 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
61 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
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 ...
1
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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.
1
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1answer
95 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 $p_1,p_2,\ldots,...
0
votes
1answer
79 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. http://...
1
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1answer
56 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
62 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
32 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 http://mathworld.wolfram.com/Euler-...
0
votes
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.
0
votes
1answer
29 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
97 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 $x^2+...
-1
votes
2answers
79 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.
31
votes
0answers
515 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)} = \Big(1+\frac{...
1
vote
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 ?
1
vote
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 $a(k^*)...
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 $3,6,7,11,12,14,15,19,21,22,23,24,27,28,...
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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
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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
votes
1answer
20 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 + (y+...