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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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 ...
jozinho22's user avatar
  • 113
1 vote
0 answers
61 views

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 ...
Aniket Kumar's user avatar
1 vote
2 answers
68 views

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 ...
Kieren MacMillan's user avatar
3 votes
3 answers
126 views

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)...
youthdoo's user avatar
  • 705
3 votes
4 answers
202 views
+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 ...
Ionela Buciu's user avatar
5 votes
0 answers
182 views
+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!...
Thirdy Yabata's user avatar
-3 votes
0 answers
42 views

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 ...
Ionela Buciu's user avatar
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4 answers
79 views

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 ...
Ionela Buciu's user avatar
-1 votes
0 answers
79 views

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 ...
Allan P.'s user avatar
1 vote
3 answers
90 views

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 ...
Ionela Buciu's user avatar
1 vote
1 answer
44 views

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: $...
Ionela Buciu's user avatar
-1 votes
0 answers
45 views

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?
bpgergo's user avatar
  • 99
0 votes
1 answer
53 views

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 ...
Enkt Enktson's user avatar
1 vote
1 answer
69 views

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 $...
Shivansh Jaiswal's user avatar
2 votes
0 answers
39 views

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 $x^2 - 2$ ...
Zack Reagin's user avatar
1 vote
0 answers
85 views

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,...
Eliezer Meth's user avatar
4 votes
2 answers
146 views

$a^2+a+3b$ 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 ...
Abu Nussa's user avatar
2 votes
2 answers
162 views

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,\...
Keith Backman's user avatar
5 votes
0 answers
145 views

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. ...
Ionela Buciu's user avatar
0 votes
1 answer
71 views

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 ...
Keith Backman's user avatar
2 votes
0 answers
84 views

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 ...
UnsinkableSam's user avatar
0 votes
2 answers
180 views

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 ...
Thirdy Yabata's user avatar
6 votes
2 answers
178 views

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 ...
Thirdy Yabata's user avatar
1 vote
0 answers
66 views

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)...
1196315's user avatar
  • 31
-3 votes
1 answer
122 views

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, ...
Explisse NA's user avatar
13 votes
1 answer
2k views

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 ...
jiraffe's user avatar
  • 165
4 votes
1 answer
156 views

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 ...
Shaun's user avatar
  • 43k
1 vote
0 answers
59 views

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 ...
vvg's user avatar
  • 3,133
0 votes
0 answers
22 views

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 ...
Kevin Stefanov's user avatar
1 vote
1 answer
95 views

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 $(n,x,y)=(64,2,6)$. ...
Eric Mairene's user avatar
2 votes
1 answer
126 views

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 ...
Haseen Siddiqui's user avatar
0 votes
1 answer
82 views

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, ...
James's user avatar
  • 700
0 votes
1 answer
87 views

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} \...
Lorenz H Menke's user avatar
0 votes
1 answer
65 views

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 ...
Aatmaj's user avatar
  • 1,065
2 votes
1 answer
161 views

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 $n=5$, ...
Thirdy Yabata's user avatar
0 votes
0 answers
63 views

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 $f(m)\to\infty$ ...
user avatar
0 votes
1 answer
123 views

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}<...
Stefan Solomon's user avatar
1 vote
1 answer
75 views

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 ...
Robert J. McGehee's user avatar
0 votes
1 answer
94 views

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 = ...
Michal's user avatar
  • 117
2 votes
0 answers
213 views

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 ...
user25406's user avatar
  • 978
2 votes
1 answer
96 views

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 ...
user25406's user avatar
  • 978
1 vote
1 answer
137 views

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 $1!^3+2!^3+3!^3+…n!^3$ ...
Thirdy Yabata's user avatar
2 votes
2 answers
96 views

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=...
user avatar
2 votes
1 answer
86 views

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 ...
user avatar
1 vote
1 answer
60 views

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^...
user avatar
0 votes
2 answers
50 views

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 ...
user avatar
1 vote
0 answers
86 views

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 ...
UniformConvergence's user avatar
1 vote
1 answer
391 views

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 ...
Jose Arnaldo Bebita Dris's user avatar
2 votes
1 answer
110 views

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....
youthdoo's user avatar
  • 705
0 votes
0 answers
67 views

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 ...
Arthur Queiroz Moura's user avatar

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