Questions tagged [square-numbers]

This tag is for questions involving square numbers. A non-negative integer $n$ is called a square number if $n = k^2$ for some integer $k$. Consider using with the [elementary-number-theory] or [number-theory] tags.

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2answers
52 views

What primes $p$ satisfying $11 p+1$ to be a perfect square?

I have tried to get prime values for which 11 p+1 is a perfect square but i didn't succeed to get a solution, I have started from :$(11p+1) \mod 2=0=m^2$ this means $11p+1 \bmod 2=0 $ implies $11 p\...
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3answers
240 views

Find all integers $m,\ n$ such that $m^2+4n$ and $n^2+4m$ are both squares. [closed]

Find all integers $m,\ n$ such that both $m^2+4n$ and $n^2+4m$ are perfect squares. I cannot solve this, except the cases when $m=n$.
3
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0answers
51 views

Are there infinite square free integers $4x+1, 4x+2, 4x+3$

Clearly, $4$ divides $(x)(x+1)(x+2)(x+3)$ Likewise, if a prime $p^2$ divides $x$, $p^2$ does not divide $(x+p)(x+2p)\dots(x+[p-1]p)$. Does it follow that there are an infinite number of instances $...
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2answers
57 views

Close to the integer squares ?? $ f(x) = \sum_{n=1}^{\infty} g(n)^{-x} ,f(4 n) = \zeta(8 n) $

Consider $g(n)$ an integer function mapping positive integers to positive integers. Also $g(n)$ is strictly nondecreasing. Thus $g(n+1) > g(n) - 1$. Define $f(x)$ for $x>1$ as $$ f(x) = \sum_{...
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2answers
45 views

Square Numbers general methods

Let this equation be given: $$a^x+a^y=b^2$$ With which method can one determine if there are infinitely many or no solutions for a random number $a$ so that $b^2$ is a square number. $a, b, x$ and $...
5
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3answers
166 views

Why is $\underbrace{444\dots44}_{2n} + \underbrace{888\dots88}_{n} + 4$ never a perfect square?

In this question, the questioner asked to prove that $$f(n)=\underbrace{444\dots44}_{2n} + \underbrace{888\dots88}_{n} + 4$$ is a perfect square for all $n\in\mathbb N$. However, I was not able to ...
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1answer
107 views

444…44+2×888…88+4 is perfect square?

Prove that $\underbrace{444\dots44}_{2n} + 2×\underbrace{888\dots88}_{n} + 4$ is a perfect square For me I dunno how to do this, I could only managed to notice that 4 is perfect square. Please help
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2answers
110 views

Product of numbers in a set

What is the least number of elements we have to delete from the set {10, 20, 30, 40, 50, 60, 70, 80, 90} so that the product of the elements remaining in the set is a perfect square?
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7answers
4k views

Can any number of squares sum to a square?

Suppose $$a^2 = \sum_{i=1}^k b_i^2$$ where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct). Can any positive integer be the value of $k$? The reason I am ...
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2answers
136 views

Proving that a list of perfect square numbers is complete

Well, I have a number $n$ that is given by: $$n=1+12x^2\left(1+x\right)\tag1$$ I want to find $x\in\mathbb{Z}$ such that $n$ is a perfect square. I found the following solutions: $$\left(x,n\...
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2answers
110 views

$\frac{1}{5}\big((4 + \sqrt{15})^{2n} + (4 - \sqrt{15})^{2n} + 8\big)$ is the sum of 3 consecutive squares

Prove that for every positive integer $n$, the number $$ \large \frac{(4 + \sqrt{15})^{2n} + (4 - \sqrt{15})^{2n} + 8}{5} $$ can be expressed as a sum of squares of three consecutive integers. ...
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1answer
64 views

When does a certain number is a perfect square

I've the following number: $$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$ Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
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1answer
58 views

When is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square for $x\ge2$ and $x\in\mathbb{N}$

I've the following number: $$81 + 60 x (1 + x) (-2 + 5 x)$$ For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square?
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1answer
138 views

For what value of $x$ is the following number a perfect square

I've the following number: $$1+12x^2(1+x)$$ For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $1+12x^2(1+x)$ a perfect square?
1
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1answer
53 views

Number of spread numbers in the field $\mathbb{F}_{p}$

I'm reading a book defining a "spread number" in a prime field a number $s \in \mathbb{F}_{p}$ verifying $s(1-s)$ being a square number. Then, the book says without proof that: if $p = 1 (\mod 4)$ ...
2
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1answer
78 views

Sum of powers is a perfect square

Show that $ \sum_{k=1}^{2000} {2^k+7^k+9^k}$ îs a perfect square. I tried grouping terms or evaluating the geometric progressions...but without success. I got $$\frac{48∗2^{2000}+28∗7^{2000}+27∗9^{...
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4answers
124 views

Prove that $\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}\sim\frac1{2n}$

I would like a proof of the asymptotic relationship $$\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}\sim\frac1{2n}$$ without assuming that the sum is a Riemann sum. This problem arose from Question ...
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1answer
32 views

Special cases of solving a square root [closed]

Are there cases where solving a square root, which you can move the coefficient inside the √ to yield a rational answer, but cannot yield one if you leave the coefficient on the outside. I imagine ...
1
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1answer
104 views

Find all $n$ natural numbers such that : $n^{2}-3$ divisible by a perfect square $m>1$

Problem : Find all $n$ natural numbers such that : $k=n^{2}-3$ divisible by a perfect square $m>1$ I'm going to find the smallest number then find all this numbers I was tired many time of $n$...
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1answer
230 views

Is there a way to check if an integer is a square?

Is there a way to check if a number is square number? For example, we know that $4$ is a square number because $2^2=2$ and $9$ is a square number because $3^2=9$. But for example $5$ is not a square ...
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2answers
263 views

Prove square root of 16 is rational [closed]

I am student and i understand that the square root of any perfect square is a rational number but i'am trying to prove it (e.g for 16).
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5answers
53 views

Using the fundamental theorem of arithmetics

I’ve got a problem that I’m not quite sure how to solve. I can see the reasoning behind the problem, but I’m not sure how to apply the theorem. Suppose that $a$, $b$, and $c$ are integers, and $a^...
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3answers
45 views

Prove that when $n$ is square free, then $a^2b = a^2c \text{ mod }n$ implies that $ab = ac \text{ mod } n$

I am trying to prove that when $n$ is a square-free composite number ($n = p_1\dots p_r$ where $p_i$ are primes), then $a^2b = a^2c \text{ mod }n$ implies that $ab = ac \text{ mod } n$. I have tried ...
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2answers
70 views

Selection of M square numbers to make a square

I came across a question during an exam of computational mathematics to which I could think of no solution other than a brute force approach. Given the squares of first 9 natural numbers, i.e $1,4,...
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1answer
44 views

(Continue) About square numbers

It's the question from these threads: Asking for suggestions about square numbers (Again) About square numbers Don suggested my trying to explain it more clearly because my scribbles were too ...
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1answer
37 views

(Again) About square numbers [closed]

Here I am again, trying to organize my thought. Thank you Don for encouraging me yesterday. (From this thread Asking for suggestions about square numbers) I'll try to explain as much as I can, but ...
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0answers
52 views

Asking for suggestions about square numbers

Please let me excuse first. It is embarrassing but I know nothing about math; but when I played with the square numbers with my calculator while slacking of my paper work yesterday (like, press 4 then ...
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1answer
25 views

smallest perimeter of compound shape

If you have a compound shape made of three unique squares with fixed sizes, what is the smallest possible perimeter for that shape? assuming no overlaps.
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2answers
187 views

Demonstrate that there are no perfect squares ending with $8$ [duplicate]

A number n will always end in some digit of the set {$0,1,2,3,4,5,6,7,8,9$}. The last digit of $n^2$ is the last digit of its last squared digit. Like this: $$\ldots 0^2 = \ldots 0$$ $$\ldots 1^2 = \...
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2answers
31 views

Perfect squares relantionship [closed]

Find all natural numbers $\overline{xyzt}=10^3x+10^2y+10z+t$ who satisfies the following condition $$ \sqrt{x}+2\sqrt{y}+3\sqrt{\overline{zy}}+4\sqrt{z}=t^2. $$
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0answers
24 views

Using the 'reductio ad absurdum' to prove the non-existence of two positive integers given a condition

I found this problem in the 'logic' section of my discrete mathematics textbook (I am currently a freshman majoring in Computer Science). It is a demonstration problem and I feel like I might have ...
2
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0answers
82 views

Finding squares that add to a certain sum

I have been looking for an algebraic way to solve for 2 squares when given a sum, but I got nothing so far. For example: $$y^2 + x^2 = 9797$$ I thought that the solution would have to do something ...
3
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1answer
55 views

Numeric functions

Let $n$ and $k$ be positive integers. A function $f$:{$1, 2, 3, 4, ... kn$} --> {$1, ... 5$} is said to be good if $f (j + k) - f (j)$ is a multiple of $k$ for all $j = 1.2, ..., kn - k$ a) Prove ...
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2answers
70 views

How can be proved $\sqrt{n^8+2\cdot 7^{m}+4}$ is irrational?

How can be proved $\sqrt{n^8+2\cdot 7^{m}+4}$ is irrational, where $n,m \in \mathbb{N}$. I tried to write: $$a^2-n^8=2(7^m+2)$$ and to try finding the last digit in the left and in the right side. ...
2
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2answers
138 views

Pairs of perfect squares

Two perfect squares are said to be friendly if one is obtained from the other by adding the digit 1 on the left. For example, $1225 = 35 ^ 2$ and $225 = 15 ^ 2$ are friendly. Prove that there are in ...
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2answers
88 views

How do I solve this equation for integer solutions? [closed]

How do I find integer solutions for the following equation? $x^2 + 2y = 1$
1
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1answer
35 views

Pattern in the list of square-triangular numbers.

I was just reading about Square Triangular numbers on its Wikipedia page (https://en.wikipedia.org/wiki/Square_triangular_number) and noticed a peculiar pattern in them. Given below is the list of ...
7
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1answer
144 views

A Diophantine Equation related to arithmetic progression: $T_n=a^n+b^n+c^n$.

I'm working on a problem that for complex $a,b,c$, when will these four numbers $a+b+c, a^2+b^2+c^2, a^3+b^3+c^3, a^4+b^4+c^4$ becomes an arithmetic progression with integer values. If we let this ...
0
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1answer
17 views

why value of function with sqrt changes

I have this function (1) $f(x) = {\sqrt (x+1) - \sqrt x}$ then I make it (by multiply with : $\frac{\sqrt (x+1)+\sqrt x}{\sqrt (x+1)+\sqrt x}$) into: (2) $f(x) = \frac{1}{\sqrt (x+1)+\sqrt x}$ the ...
1
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1answer
45 views

If $T_n$ is the $n$-th triangular number, there are an infinite number of $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2 $ for all $n$.

If $T_n$ is the $n$-th triangular number, show that there are an infinite number of positive integers $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2 $ for all $n$. This is inspired by an article in ...
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4answers
242 views

Two integers whose product is square [closed]

I am trying to prove the following statement. Let $d = \gcd(x_1,x_2)$. If $x_1 x_2$ is a square, then ($x_1 = d M^{2}$ and $x_2 = dN^2$) or ($x_1 = -d M^{2}$ and $x_2 = -dN^2$), where $\gcd(N,M) =1$. ...
2
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1answer
58 views

For what values of $a$ does $b\in\mathbb{N}$?

I've the following function: $$b=\frac{1}{2}-\frac{1}{r-2}+\sqrt{\frac{1}{4}-\frac{r-3}{(r-2)^2}+\frac{a(1+a)}{r-2}+\frac{a(a^2-1)}{3}}$$ I know that: $\text{a}\ge1$ and $a\in\mathbb{N}$; $\text{r}\...
2
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3answers
37 views

Roots of x inside square root

In finding the roots of $ \sqrt{6-4x-x^2}=x+4$ I get that the roots of $x$ are $-5$ and $-1$. However first I need to take into account that: $x+4 \ge 0$ and $6-4x-x^2 \ge 0$. Considering the ...
1
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2answers
208 views

Sum of squares of 3 consecutive numbers is not a perfect square

I'm trying to show that $(n-1)^2+n^2+(n+1)^2=a^2$ does not have a solution for $n,a\in \Bbb N$. I've written $(n-1)^2+n^2+(n+1)^2=3n^2+2$, so what I need to show is that $3n^2+2$ cannot be a perfect ...
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2answers
115 views

Find all $n$ for which $3n^2+3n+1$ is a perfect square.

Find all natural numbers $n$ for which $3n^2+3n+1$ is a perfect square. I used discriminant method but failed. Then I found upper and lower bounds of this expression: Lower:$(n+1)^2$ Upper:$(2n)^2$ ...
20
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2answers
217 views

Sequences where $\sum\limits_{n=k}^{\infty}{a_n}=\sum\limits_{n=k}^{\infty}{a_n^2}$

I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form. I then wondered ...
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3answers
75 views

Determine the integers $m$ such that $1+2+3+…m=$ a perfect square

I went like this $ \frac{m(m+1)}{2} = k^2 ; m^2+m = 2k^2$ From here I noticed that m is even that is $m =2q$ Substituting it will give $ 2q^2+q = k^2$ From here I got no where.. Then I ...
3
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2answers
95 views

Positive integer solutions to $ p+q^{n}=x^2$

Consider two prime numbers $p$ and $q$ such that $$ p+q^2=r^2, $$ and $r\in\mathbb N$. It is not difficult to figure out that for any $n\in\mathbb N$ and $x\in\mathbb N$ there are no solutions of $$ ...
1
vote
1answer
85 views

Sum of digits of squares of a number and its reverse

I have observed the following: Let $ab$ be a two digit number where $a$ and $b$ represent digits, and $ba$ be a two digit number with its digits in reverse order as $ab$. If the sum of the digits ...
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4answers
50 views

Determining number of integer solutions for expression of perfect square

How many positive integer values of n are there such that $2^n + 7^n$ is a perfect square? I am not sure how to approach this question given that there are two different bases 2 and 7

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