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.
1,280
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Is there a function that can cancel a square of a number, preserving its initial value?
When we have the square function defined by :
$y = x²$
This implies $x = \sqrt{y}$ or $x = - \sqrt{y}$, because the function $f(x) = \sqrt{x}$ can have only one image, and does return only a positive ...
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Is it possible for three distinct positive integers to exist such that sum and difference of every two is a perfect square?
I have been struggling with this problem for quite some time now but i haven't been able to find a solution yet.
My first approach was to try and make an Euler's Brick type form but that didn't really ...
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If $x>y\ge1$ are integers, with $x$ even and $y$ odd, and $x^4+4x^3y-4xy^3-4y^4$ is a square, must $y \mid x$?
The title says it all: Let $x$ and $y$ be integers, with $x$ even and $y$ odd, such that $x > y \ge 1$ and
$$x^4+4x^3y-4xy^3-4y^4 = w^2$$
for some integer $w$. Does this force $y \mid x$?
Brute ...
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3
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Prove $a=b$ if $4p^{2a}+4p^b+1$ is a perfect square
Let $p$ be an odd prime, and $a$, $b$ be two positive integers such that $2a>b$. If $4p^{2a}+4p^b+1$ is a perfect square, prove that $a=b$.
If $b<a$, then $(2p^a)^2<4p^{2a}+4p^b+1<(2p^a+1)...
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+100
Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?
Check if there are nonzero natural numbers $n,x,y$ such that:
$$\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}. $$Thank you in advance!
My ideas
So we can simply show that $4n+5,5n+1,9n+4$ are ...
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+50
Can $!1+!2+!3+\cdots+!n$ be a perfect power?
Can $!1+!2+!3+\cdots+!n$ be a perfect power if $n\geq3$?
Note that $!n$ is a subfactorial.
I do know that $1!+2!+3+\cdots+n!$ is only a perfect power if $n=1, 3$, since when $n\geq9, 1!+2!+3!+\cdots+9!...
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How can we show that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares?
I saw a lot of problems that assume this: $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares. I wonder how can we demonstrate it because i saw a lot of people using ...
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Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.
QUESTION
Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.
MY IDEA
I decided to write it as:
$2x^2+2y^2-4xy+x^2+x-2=0={(x\sqrt{2}+y\sqrt{2})}^2+x^2+x-2$
I was thinking of ...
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Square Formula I came up with. Criticize me! [closed]
You can apply this to squares from between $10^2$ up until $99^2$
Multiply the ones place of the to-be-squared number by itself to get the ones place of the solution, add the ones place to itself to ...
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Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$
Let $a$, $b$ and $c$ be nonzero natural numbers. Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$.
My ideas
For those numebrs to ...
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Others abbreviated calculation formulas...
I started sloving an algebric problem and i wonder if we can write $x^2+y^2+z^2$ or $a^2+b^2+c^2+2a+2b+2c+3$ as a product of terms.
By product of terms i think of writing does terms as a product:
Ex: $...
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prove any 5 digit number containing the following digits is not perfect square: 1, 1, 4, 6, 9 [closed]
60 different 5 digit number can be created from these digits: 1, 1, 4, 6, 9
set(permutations([1, 1, 4, 6, 9]))
how to prove none of them is a perfect square?
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Number theory on multiples of squares [duplicate]
If we take the numbers $ 48 $, $ 49 $ and $ 50 $, we can see that they are all consecutive integers and multiples of squares ($48$ is multiple of $2^2$, $49$ is multiple of $7^2$ and $50$ is multiple ...
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How to find values of the given $4n^2 - 5n + 16$ for which the function is a perfect square?
Problem: Given a positive integer $m$, which can be written as $m^2 = 4n^2 - 5n + 16$, where $n$ is an integer of any sign, find the maximum value of $|m - n|$.
For this, I assumed $|m - n| = k$ then
$...
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Find the $n^{th}$ number, $p$, where $p$ is a prime number and $x^2-2$ is evenly divisible by $p$, where $x$ is an integer
I am trying to complete a coding challenge where, given a number, $n$, I need to create a program to output the $n^{th}$ prime number, $p$, for which there exists an integer, $x$, such that $x^2 - 2$ ...
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Calculating square roots using perfect squares
I recently discovered a way to quickly calculate perfect squares (that are close to a prior-known perfect square), then extrapolated from that a method to mentally calculate the square root of numbers,...
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$a^2+a+3b$ and $b^2+b+3a$ both perfect square
Find all posssible $(a,b)$ where $a$ and $b$ natural number (positive integer) such that $a^2+a+3b$ and $b^2+b+3a$ both perfect square
so, i try to assume that
$a^2+a+3b=(a+m)^2$ with $m$ natural ...
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2
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162
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Can the sum of squares of odd primes equal the square of an odd prime?
In this question it was shown that collections of $8k+1$ odd squares can be found that sum to an odd square. In his answer, Denis Shatrov provided an algorithm by which $8k$ odd numbers $\{a_1,a_2,\...
- 6,987
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Showing $x+y>z$, where $x=\sqrt{10}+\sqrt{26}$, $y=\sqrt{17}+\sqrt{37}$, and $z=\sqrt{323}$. Is my idea corect?
Let the numbers be $x=\sqrt{10}+\sqrt{26}$, $y=\sqrt{17}+\sqrt{37}$, and $z=\sqrt{323}$. Show that $x+y>z$.
MY IDEA
So I thought if i write every number in a interval of 2 perfect square numbers.
...
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1
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Multiple odd squares sum to an odd square
Any number of appropriately chosen squares can sum to a square. However, this is not the case when the square addends and the square sum are all required to be odd. Since all odd squares are $\equiv 1 ...
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Concatenation of square numbers is a square?
Just a curiosity of mine.
If I define the $n^{th}$ concatenation number (denoted $Q_n$) to be the the concatenation of digits from the $1^{st}$ square number to the $n^{th}$, can $Q_n$ ever be square ...
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180
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Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal? [closed]
Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal?
I know that $1!+2!+3!+...n!$ is never a perfect square if $n\geq5$, since the last digit of the sum is $3$, but I don't know if ...
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Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power?
Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power?
I noticed that $(7!+6!+5!+4!+3!+2!+1!)+16=77^2$ is a perfect power, and I don’t know if that is the largest ...
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66
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How to find a perfect natural power number in vicinity of an arbitrary number? [closed]
If I pick a natural number $n \in \mathbb{N}$ such that $n-x = a$ is a perfect natural power and $n+y= b$ is also a perfect natural power, then are there any methods or conditions to find $minimum(x,y)...
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Can the sequence 2+4+6+8+10+...+2n at some point be equal to a perfect square? [closed]
It's a question I've been asking myself for a long time, without finding the answer.
There are also variants, such as the sequence 2+6+8+10+12+14+... which is in fact the same, but without the term 4, ...
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Can a number written only with zeros, threes, fives, sevens and eights be a perfect square?
Can a number written only with zeros, threes, fives, sevens, and eights be a perfect square?
I've tried to look at the residues modulo $3$ and $9$ with the sum of the numbers, but I've gotten no ...
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Numbers such that $(\overline{a_1\dots a_n})^2=\overline{x_1\dots x_m}$ and $(\overline{a_n\dots a_1})^2=\overline{x_m\dots x_1}$.
Recently, I had the pleasure of finding out that
$$13^2=169\quad\text{and}\quad 31^2=961.$$
It had me wondering . . .
The Question:
What pairs of distinct natural numbers $r,s$ have decimal ...
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The maximum weight in a weighted sum of Fibonacci squares representation of a positive integer
Theorem. Every positive integer can be represented as a weighted sum of Fibonacci squares.
Proof. Start with a postive integer $N$. Obtain its unique sum of non-consecutive Fibonacci numbers ...
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Specialized algorithms for edge cases of binary arithmetic
I have several mathematical operations on binary numbers that are special cases of more general arithmetic operations. I am wondering whether there exist more specialized algorithms purpose-made for ...
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Property $x^y = (x+y)^2$ [closed]
Solve for all $n \in \mathbb{N}$ such that:
$$n=x^y=\left(x+y\right)^2$$
where $x,y \in \mathbb{Z+}$, and $x$ or $y$ is a prime number.
this is the question, I only got a triple $(n,x,y)=(64,2,6)$.
...
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How to estimate a square root of a decimal number?
The way I estimate square roots, is by finding the closest lowest perfect square, then adding decimals to the number to determine the estimation. How do I estimate the square root of a number with ...
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How to compute the square root of a polynomial modulo another polynomial?
What is a method to find the polynomial square root modulo another polynomial, i.e. find a polynomial $r(x)$ such that
$$r(x)^2 \mod{x^3+3x+1} = -453600x^2 - 160650x - 4725$$
?
From various sources, ...
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Evaluation sum and its asymptote $\sum_{s=1}^{N} \sum_{t=1}^{N} \left[{\sqrt{4t-s^2} \in \mathbb{Z}}\right]$
I am working on the evaluation of $$S \left({N}\right) = \sum_{s=1}^{N} \sum_{t=1}^{N} \left[{\sqrt{4t-s^2} \in \mathbb{Z}}\right]$$ and its asymptotic expansion where $N \ge 1$. Here $\sqrt{4t-s^2} \...
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Perfect square in the sequence formed by consecutive numbers
I recently saw a question Is there any perfect square in the sequence $12,123,1234,12345,...$?
This led to thinking about a new question.
Consider sequence https://oeis.org/A057137 that is the ...
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Is $1^{1!}+2^{2!}+3^{3!}+\dots n^{n!}$ a perfect square?
Is $1^{1!}+2^{2!}+3^{3!}+\dots n^{n!}$ a perfect square?
Obviously, $1$ is a perfect square. When $n=2$, the sum is $5$, which is not a perfect square. When $n=3,4$, the sum is $2\pmod{4}$. For $n=5$, ...
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Help me (dis)prove a conjecture about square numbers
Let $S=k*m+p$ be any number where $k,m\in\mathbb{N}$ and $p$ is a prime number.
My claim is the following:
There exists some nondecreasing function $f:\mathbb{N}\mapsto\mathbb{N}$ with $f(m)\to\infty$ ...
user1114342
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Find the sequence of integers that satisfies a recursive condition.
We have a natural number $n\geq4$ and the set A= {$a_{1},a_{2},...,a_{n}$} with natural numbers. We know $a_{k}+k\sqrt{a_{k+1}} \in A $ for every $k \in \{{1,2,...,n-2}\}$. We also know that $a_{1}<...
- 1,161
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1
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Quadratic residues that are also squares themselves
I have a question about number bases and square numbers. When considering quadratic residues in a given modulus (without eliminating residues that are not relatively prime to the modulus), it is ...
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1
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How to prove that specific value (...5) is infinite in 10-adic
I have watched: Veritasium
The idea behind is that there exist value that satisfies: $n^2 = n$.
That value is its own square.
We can create this value by:
$$
5^2 = 25
$$
$$
25 ^ 2 = 625
$$
$$
625^2 = ...
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Does using pre-computed squares speed up significantly the calculation of factorial $n!$?
There are many different methods that tried to improve the calculation of $n!$. Few of them managed to halve the number of mulitplications. One of those methods is the basis to completely remove the ...
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Why does a sum of factorials behave differently from single factorials?
The Brocard Problem shows three factorials that can be expressed as $n!+1=m^2$ or equivalently $n!=k(k+2)$. So any time a single factorial can be put in the form of $k(k+2)$, it will belong to Brocard ...
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Is $1!^3+2!^3+3!^3+…n!^3$ a perfect square when $n>3$?
I noticed that $1!^3=1$ and $1!^3+2!^3=9$ are both perfect squares.
$1!^3+2!^3+3!^3=225=15^2$ is also a perfect square and even if $n>5$, it does end on $9$, so I think that $1!^3+2!^3+3!^3+…n!^3$ ...
2
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2
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Find all two-digit prime number pairs $p$ and $q$, for which $pq+1$ is a perfect square.
Find all two-digit prime number pairs p and q, for which $p\cdot q+1$ is a perfect square.
MY IDEAS
Because p and q are two digit prime numbers, clearly, they are odd numbers, so let $p=2a+1$ and $q=...
user1104319
2
votes
1
answer
86
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help Find the real numbers $x,y,z$ such that $(x+y)\cdot x= \frac{a^{2}}{2}$ and $(z+y)\cdot z= \frac{b^{2}}{2}$ and $x+y+z= \frac{a+b}{\sqrt{2}}$
Find the real numbers $x,y,z$ such that $(x+y)\cdot x= \frac{a^{2}}{2}$ and $(z+y)\cdot z= \frac{b^{2}}{2}$ and $x+y+z= \frac{a+b}{\sqrt{2}}$ where $a$ and $b$ are positive real numbers
.
MY IDEAS
...
user1104319
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1
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Solve $x, y \in \mathbb{R}$ given $x(12-x)+y(16-y)=100$.
Solve the equation in $\mathbb{R}$
$$x(12-x)+y(16-y)=100$$
MY IDEAS:
\begin{aligned}
& x(12-x)+y(16-y)=100 \\
& 12 x-x^2+y^{16}-y^2=100 \\
& 12 x+16 y-x^2-y^2=100 \\
& 12 x+16 y-x^2-y^...
user1104319
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2
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Determine the numbers a and b that check the equality $\sqrt{\overline{aba}}=(a+b-1)\cdot \sqrt{a+b}$
Determine the numbers a and b that check the equality
$\sqrt{\overline{aba}}=(a+b-1)\cdot \sqrt{a+b}$
MY IDEAS
I thought of decomposing $\overline{aba}$ as $101\cdot a + 10\cdot b$
Then i thought that ...
user1104319
1
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0
answers
86
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An elementary proof that there are no 4 squares in arithmetic progression
My problem.
I am trying to fill in the details in @lhf 's outline of an elementary proof (mentioned in Dickson's History of the theory of numbers) that there are no four Squares in arithmetic ...
1
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1
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Under what conditions is it true that $m^2 - a$ is not a square if and only if $(m - 1)^2 < m^2 - a < m^2$, where $a>0$?
Preamble: The present inquiry is an offshoot of this MSE question from February 21, 2019, and ultimately, Theorem III.2, page 2 from a paper submitted to a conference organized by De La Salle ...
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1
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For how many numbers $1\le k\le n$ does there exist $a$, $b\in\Bbb Z$, so that $k=a^2+b^2$?
For how many numbers $1\le k\le n$ does there exist $a$, $b\in\Bbb Z$, so that $k=a^2+b^2$? Give an approximation, focus more on the lower bound.
Let the number be $f(n)$. First, I know the following....
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Which integers are the sum of $k$ positive squares?
For each $k$, is there something we can say about each positive integer being able to be written as a sum of $k$ positive squares? What is known for $k > 5$? What about if we only care about ...
