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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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How to generalise the argument in Chap. 1 in Baby Rudin to show that these sets $A$ and $B$ have no largest and smallest elements, respectively?

Let $n$ be a positive integer that is NOT a perfect square, and let the sets $A$ and $B$ be defined as follows: $$ A := \left\{ p \in \mathbb{Q} \colon p > 0, p^2 < n \right\} $$ and $$ B := \...
Saaqib Mahmood's user avatar
2 votes
0 answers
65 views

Except for $1$, are there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$?

Other than $1$, is there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$? This is sequence A098129. We certainly know that this is not a ...
Thirdy Yabata's user avatar
2 votes
3 answers
86 views

Find the natural numbers $x,y$ if $\sqrt{x-1}+\sqrt{x+2023}=y$

Question Find the natural numbers $x,y$ if $\sqrt{x-1}+\sqrt{x+2023}=y$ my idea $\sqrt{A}+\sqrt{B}=N$ where we noted $A=x-1$ and $B=x+2023$ and $N=y$ $\sqrt{A}=N-\sqrt{B}|^2$ $A=N^2+B-2N\sqrt{B}$ $\...
IONELA BUCIU's user avatar
  • 1,393
0 votes
0 answers
25 views

Relative distribution of primes among consecutive squares

For each prime $p,$ define $n_p>0; {n_p}^2$ is the greatest square integer $\leq p,$ so that ${n_p}^2 \leq p < (n_p + 1)^2.$ [Now note that $(n_p + 1)^2-1-{n_p}^2 = 2n_p $]. Is $S:= \left\{ \...
Adam Rubinson's user avatar
2 votes
1 answer
60 views

Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power?

Is $k=11$ the largest value of $k$ such that $(\lfloor\sqrt{1!+2!+3!+\cdots+k!}\rfloor+1)^{2} - (1!+2!+3!+\cdots+k!)$ is a perfect power? While I playing on my calculator, I observed the following: $(\...
Thirdy Yabata's user avatar
1 vote
1 answer
49 views

Is it true that for any positive integer $n$, there exists an integer $x$ where there are at least $n$ primes between $x^2$ and $(x+1)^2$

Am I correct that this follows directly from two observations: (1) The sum of the reciprocals of primes diverges. (2) The sum of the reciprocals of squares converges Here's my thinking: If there ...
Larry Freeman's user avatar
-1 votes
2 answers
107 views

Find the perfect squares of four digits whose square root is the sum of the numbers obtained if we separate the first two digits from the last two

the question We are asking for the perfect squares of four digits whose square root is the sum of the numbers obtained if we separate the first two digits from the last two the idea let the number be $...
IONELA BUCIU's user avatar
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0 votes
1 answer
69 views

Show that if $x + y + z = xy + xz + yz$ then $x ^ 2 * (1 - y ^ 2) + y ^ 2 * (1 - z ^ 2) + z ^ 2 * ( 1 - x ^ 2) = 2(x + y + z)(xyz - 1)$

the question Show that if $x + y + z = xy + xz + yz$ then $x ^ 2 * (1 - y ^ 2) + y ^ 2 * (1 - z ^ 2) + z ^ 2 * ( 1 - x ^ 2) = 2(x + y + z)(xyz - 1)$ the idea First of all i though of breaking all the ...
IONELA BUCIU's user avatar
  • 1,393
2 votes
2 answers
85 views

How to prove the existence of this progression? Struggling with a BDMO problem.

This problem is from Bangladesh Mathematical Olympiad $2023$, The problem statement is as follows- Prove that there is sequence of $2023$ distinct positive integers such that the sum of the squares of ...
Sonia Sultana's user avatar
6 votes
0 answers
79 views

how many natural numbers require at least 6 terms to express as the sum of distinct squares?

I wrote a computer program as an exercise in dynamic programming. It finds the minimum number of distinct squares which sum to some positive target integer n. I found something interesting and would ...
Simon Goater's user avatar
2 votes
3 answers
101 views

Find $n\in N$ for which $2*[\frac{1^2}{2}]+2^2*[\frac{2^2}{3}]+...+2^n*[\frac{n^2}{ n+1}]$

Question Find $n\in N$ for which $$2 \times \left[\frac{1^2}{2}\right] + 2^2 \times \left[\frac{2^2}{3}\right] + ... + 2^n \times \left[\frac{n^2}{n+1}\right] = 2^{2025} \times 2022 + 4$$ where $[a]$ ...
IONELA BUCIU's user avatar
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2 votes
3 answers
174 views

Can $4\cdots41$ (with odd number of $4$s) be a Square Number? [closed]

Consider a number in its decimal representation that begins with an odd number of consecutive digits of 4, followed by a single digit of 1. An example of such a number would be 41, 4441, or any ...
Wismar Günther's user avatar
7 votes
3 answers
202 views

Determine all the integers $x$ that have the property that $\sqrt{x^2+7x+21}$ is a rational number.

the question Determine all the integers $x$ that have the property that $\sqrt{x^2+7x+21}$ is a rational number. the idea The number would be rational only if $x^2+7x+21$ would be a square number ...
IONELA BUCIU's user avatar
  • 1,393
2 votes
1 answer
87 views

show that an expression $E(x,y)$ does not depend on x

the question Let $a,b,d,e$ be real numbers with $a>d$ and $c=a^2+b^2, f=d^2+e^2, m=\frac{b-e}{a-d} , n=\frac{bd-ae}{a-d}$. Show that if $x\in [-a,-d] $ and $y=mx+n$, then $$E(x,y)=\sqrt{x^2+y^2+2ax+...
IONELA BUCIU's user avatar
  • 1,393
5 votes
3 answers
179 views

Find $a$ for which $(a-3)(a-7)$ is a perfect square

The only solutions seem to be 3 and 7 but I can't prove that there are no others. Context: Find every value for integer a, for which $x^2-(a+5)x+5a+1$ expression can be factored as $(x+b)(x+c)$ where ...
Otar Natsvaladze's user avatar
0 votes
1 answer
108 views

Exists a perfect square that divided by 6 equals a prime number? [closed]

I recently took a test and a question came up that asked the question: ...
Max Heppelmann's user avatar
1 vote
1 answer
64 views

Sequence of squares which can't be written as the sum of a smaller non-zero square and twice a triangular number

Are there infinitely many squares which cannot be written as the sum of a smaller non-zero square and twice a triangular number? In other words, is the list given at https://oeis.org/A230312 infinite? ...
Ok-Virus2237's user avatar
3 votes
2 answers
192 views

Writing 2024 as the sum of 3 and 4 squares

I'm currently taking a course in number theory and we've just seen that any number can be written as the sum of the 4 squares, and that numbers can be written as the sum of 3 if they aren't of a ...
Skark123's user avatar
  • 107
2 votes
2 answers
123 views

How to check if something is perfect square or not.

While finding eigenvalues of a particular matrix, I end up with the following: $\sqrt{p^{2\alpha}-4p^{\alpha}+8p^{\alpha-1}+4}$, where $p$ is an odd prime and $\alpha \geq 1$. The next step is to ...
Akhil P's user avatar
  • 21
4 votes
3 answers
177 views

Determine the real numbers $a, b, c, d \in [1,3]$, knowing that the relation $(a + b + c + d)^2 = 3(a^2 + b^2 +c^2 + d^2)$.

the question Determine the real numbers $a, b, c, d \in [1,3]$, knowing that the relation $(a + b + c + d)^2 = 3(a^2 + b^2 +c^2 + d^2)$. my idea $(a+b+c+d)^2=a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd)$ $=&...
IONELA BUCIU's user avatar
  • 1,393
3 votes
1 answer
59 views

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

the question Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational. the idea A radical is rational only if the number below ...
IONELA BUCIU's user avatar
  • 1,393
0 votes
4 answers
88 views

solution-verification | Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$.

the question Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$. the idea for the number to be rational both members have to be rational (*...
IONELA BUCIU's user avatar
  • 1,393
0 votes
1 answer
33 views

solution verification | Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$. [duplicate]

The questions Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$. my idea We know that $x^2+2x+1=(x+1)^2$ $x^2+8x+16=(x+4)^2$ which ...
IONELA BUCIU's user avatar
  • 1,393
1 vote
1 answer
64 views

Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?

The questions asks: Define a formula to yield the Kth N for which there exists an integer X less than or equal to N for which the sum of the integers from 0 to X (inclusive) is equal to the sum of the ...
BlueInfinite1729's user avatar
7 votes
0 answers
132 views

Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)?

Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)? I noticed that while memorizing all perfect cubes from $1$ to $10^{6}$, $357911=71^...
Thirdy Yabata's user avatar
0 votes
3 answers
62 views

Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs

the question Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs. my idea I tried grouping them in a whole perfect square or I tried using formulas such ...
user avatar
3 votes
2 answers
90 views

Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$

the question Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $$5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$$ my idea I tried grouping them in a ...
IONELA BUCIU's user avatar
  • 1,393
1 vote
1 answer
167 views

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

The question Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$. My idea $a^2+3b^2=25n$ By using modular arithmetic( a perfect square can be only ...
IONELA BUCIU's user avatar
  • 1,393
1 vote
2 answers
84 views

Determine numbers for which $\sqrt{ \frac{\overline{ab}^2 + \overline{cd}^2}{2}} = \frac{ \overline{ab} + \overline{cd}}{ 2} +1$.

The question Determine the pairs of numbers of the form $\overline{ab} , \overline{cd}$ (numbers with 2 digits in base 10) for which the equality $$\sqrt{ \frac{\overline{ab}^2 + \overline{cd}^2}{2}} =...
IONELA BUCIU's user avatar
  • 1,393
0 votes
0 answers
45 views

solution verification || Show that there are at least 946 subsets of M for which the sum of their elements is a natural number.

Question The set is considered $$M= {\frac{1}{\sqrt{1}+\sqrt{2}}, \frac{1}{\sqrt{2}+\sqrt{3}} ,..., \frac{1}{ \sqrt{2022}+\sqrt{2023}} }$$ Show that there are at least 946 subsets of M for which the ...
IONELA BUCIU's user avatar
  • 1,393
1 vote
2 answers
90 views

Show that the number $\frac{\sqrt{2021^4+2 \cdot 2020^2-4041^2}}{505 \cdot 1011}$ is natural and a perfect cube.

question Show that the number $$\frac{\sqrt{2021^4+2 \cdot 2020^2-4041^2}}{505 \cdot 1011}$$ is natural and a perfect cube. my idea I tried writing the upper part as a perfect square so that we could ...
IONELA BUCIU's user avatar
  • 1,393
1 vote
1 answer
83 views

Find all non-negative integers (a,b) such that $k^{2}$a+b is perfect square for all integers k

The obvious solutions are ($a^{2}$,0) and (0, $b^{2}$) but I'm not sure if there are any other solutions.
MrCheese312's user avatar
1 vote
0 answers
73 views

Why can't I graph my summation equation for calculating perfect squares (The equation is correct, I tested it) [closed]

This is the equation and graph on Desmos.
david smith's user avatar
2 votes
3 answers
198 views

$N$ is not a perfect square [closed]

Prove that the number $N:=a\cdot \overline{aa}\cdot \overline{aaa}\cdot \dots \cdot\overline{aaa\dots a}$ is not a perfect square, where $a$ is a nonzero number, $\overline{aaa...a}$ contains $n$ ...
Andrey's user avatar
  • 55
1 vote
1 answer
104 views

Why does the term being squared in the Pythagorean triplet make no difference to the whole triplet?

I was recently preparing for the SOF IMO Level 2 exam. While practising for the exam, I encountered this question in my textbook about the chapter Square and Square Roots which also mentions the ...
Sambhav Khandelwal's user avatar
0 votes
0 answers
37 views

How to proof $(x^n)' = nx^{n-1} (with \enspace n \in \mathbb{R})$?

It's trivial to proove that : $$(x^n)' = nx^{n-1} \enspace (with \enspace n \in \mathbb{N}) .$$ But how to proove it with $n \in \mathbb{R}$ ? Because it works. Example : $(x^{\frac{1}{2}}) = \frac{1}{...
jozinho22's user avatar
  • 127
0 votes
1 answer
87 views

Show that $\frac{2}{(x + y + z)} \left(\frac{x ^ 3}{x ^ 2 + y ^ 2} + \frac{y ^ 3}{y ^ 2 + z ^ 2} + \frac{z ^ 3}{z ^ 2 + x ^ 2}\right) \geq 1$ [closed]

Question If $x, y, z > 0$ show that: $$\frac{2}{(x + y + z)} \left(\frac{x ^ 3}{x ^ 2 + y ^ 2} + \frac{y ^ 3}{y ^ 2 + z ^ 2} + \frac{z ^ 3}{z ^ 2 + x ^ 2}\right) \geq 1$$ My idea If we divide by $\...
IONELA BUCIU's user avatar
  • 1,393
0 votes
2 answers
177 views

Find integers solutions for which bivariate polynomial with bi-quadratic form: $4x^2y^2-4xy^2+1$, becomes a square number

Could you help me to find all integer $x$ and $y$ for which the bivariate polynomial: $$4x^2y^2-4xy^2+1$$ is a square number, i.e., it can be expressed as $z^2$ for some integer $z$? From the above, ...
Amir's user avatar
  • 5,297
0 votes
2 answers
95 views

Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime.

question Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime, for any nonzero natural number $n$. my idea $ 2^{2^{2n+1}}+2^{2^{2n}}+1 = 2^{2^{2n}}*5+1 $ i did this by giving common factor. ...
IONELA BUCIU's user avatar
  • 1,393
10 votes
1 answer
330 views

Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$?

Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$? This is sequence OEIS A000422, the concatenation of positive integers from $n$ down to $1$. If there is any perfect ...
Thirdy Yabata's user avatar
6 votes
2 answers
135 views

Determine the nonzero natural numbers $a$ and $b$ for which the number $n=\sqrt{a^{2}+6b}+\sqrt{b^{2} +6a}$ is rational.

Question Determine the nonzero natural numbers $a$ and $b$ for which the number $n=\sqrt{a^{2}+6b}+\sqrt{b^{2} +6a}$ is rational. My idea First of all because $a$ and $b$ are natural this means that ...
IONELA BUCIU's user avatar
  • 1,393
2 votes
0 answers
60 views

solution verification - determine $a, b, x$ so that $\overline{ab}^2 = \overline{xxb} $ and $\overline {ba} ^ 2 = \overline {bxx}$

question Determine the non-zero digits $a, b, x$ so that $\overline{ab}^2 = \overline{xxb} $ and $\overline {ba} ^ 2 = \overline {bxx}$. my idea We can see that $10 \leq \overline{ab} \leq 31$ for $\...
IONELA BUCIU's user avatar
  • 1,393
0 votes
0 answers
94 views

How many perfect squares can be written in the form $2{a^{2}}+3{b^{2}}$ with $a, b \in \mathbb{N}$? [duplicate]

question How many perfect squares can be written in the form $2{a^{2}}+3{b^{2}}$ with $a, b \in \mathbb{N}$? my idea I realised that the only solution is $(a,b)=(0,0)$ Let $a,b>0$ and we have to ...
IONELA BUCIU's user avatar
  • 1,393
16 votes
3 answers
721 views

Odd square as sum of $9$ distinct odd squares

I'm interested in representing odd squares as sum of $9$ distinct odd squares. So, let $n\in\mathbb{N}$ be odd and $x_1,\,x_2,\,...,\,x_9\in\mathbb{N}$ be odd and pairwise distinct. The question is, ...
summingsummer's user avatar
5 votes
2 answers
299 views

If $m$,$n$ are integers, show that $m^2-n^2$ and $m^2+n^2$ cannot both be both perfect squares. [duplicate]

As in my title, I was working on the problem that $m^2-n^2$ and $m^2+n^2$ cannot be both perfect squares. Currently I figured out that m should be odd and n should be even. The process for this part ...
Alexander Callahan's user avatar
1 vote
1 answer
79 views

How many different squares are there which are the product of six different integers from 1 to 10 inclusive?

How many different squares are there which are the product of six different integers from 1 to 10 inclusive? A similar problem, asking how many different squares are there which are the product of six ...
eee's user avatar
  • 101
11 votes
2 answers
620 views

Can $1\underbrace{44\cdots4}_{n\text{ times}}$ be a perfect power in other bases?

Can $1\underbrace{44\cdots4}_{n\text{ times}}$ be a perfect power in other bases? Inspired from this question , I know that $144$ and $1444$ are the only perfect powers in base ten, but in other bases,...
Thirdy Yabata's user avatar
0 votes
1 answer
117 views

It can be proven that $4^{2000} + 4^x + 4^{2023}$ is the square of an integer for exactly 3 integer values of x. What is the largest such value of x

It can be proven that $4^{2000} + 4^x + 4^{2023}$ is the square of an integer for exactly 3 integer values of $x$. What is the largest such value of $x$? This is a problem from a high school ...
Rattus's user avatar
  • 11
1 vote
0 answers
69 views

Is there any square harmonic divisor number greater than $1$?

A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, $n$ is a harmonic divisor number if and only if $\dfrac{nd(n)}{\sigma(n)...
Jianing Song's user avatar
  • 1,779
0 votes
0 answers
70 views

If $c\ge b\ge a$ and $c\mid a^2,b^2$, and $a$ is the minimal number satisfy $c\mid a^2$, does $a$ necessarily divide $b$? [duplicate]

Assume $a$, $b$, $c$ are positive numbers, $a \le b$ and $b \le c$. If $a^2\equiv0\pmod c$ and $b^2\equiv0\pmod c$, and $a$ is the minimal number satisfy $a^2\equiv0\pmod c$, do we always have $a\...
wddd's user avatar
  • 109

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