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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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Determine $\overline{ab}$ by $\overline{abab}+\sqrt{2-\overline{ab}+\sqrt{(\overline{ab}+1)(\overline{ab}^3-3\overline{ab}^2+2\overline{ab})+1}}=2039$

the problem Determine the natural number of the form $\overline{ab}$, written in base $10$, for which $\overline{abab}+\sqrt{2-\overline{ab}+\sqrt{(\overline{ab}+1)(\overline{ab}^3-3*\overline{ab}^2+2*...
IONELA BUCIU's user avatar
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-1 votes
2 answers
60 views

Show that $\sqrt{\frac{a^2}{b^2}+\frac{b^2}{a^2}+\frac{6b}{a}-\frac{6a}{b}+ 23}$ is a perfect square natural number.

the problem Let $a$ and $b$ be nonzero real numbers such that $a^2-b^2=3ab$. Show that $\sqrt{\frac{a^2}{b^2}+\frac{b^2}{a^2}+\frac{6b}{a}-\frac{6a}{b}+ 23}$ is a perfect square natural number. my ...
IONELA BUCIU's user avatar
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5 votes
2 answers
133 views

Proof Verification - $n$th non square number

I was trying to derive an expression for the $n$th non square number. Could someone verify if my proof is on the right track? The $n$th non-square number must lie between two consecutive square ...
MathMan's user avatar
  • 9,114
-4 votes
1 answer
49 views

Problem related to divisibility rules and digits [closed]

How do you prove that a 100-digit number that contains one hundred 0's, one hundred 1's, and one hundred 2's as its digits in a random order is not a perfect square? What about two hundred 0's, 1's, ...
Aarushi da Great's user avatar
-2 votes
1 answer
102 views

Is there a way to show that the Fibonacci subsequence $F_{6n+2}+2$ can't have any square number? [closed]

I'm investigating the Fibonacci sequence $F_{6n+2}+2$. I searched, by Maple, the first $10000$ numbers. I couldn't find any squares. I tried using quadratic residue and modularity but I got nothing ...
ThePirateKing's user avatar
0 votes
0 answers
45 views

Consecutive multiplication of natural numbers problem [duplicate]

Prove that the product of any three consecutive natural numbers is not a perfect square. If there were four numbers, I know how to solve the problem, as I would somehow mention the number $ n(n+1)(n+2)...
user avatar
0 votes
1 answer
71 views

prove that $n - \phi(n)$ is a square where $n$ is an even perfect number

where even perfect numbers are of the form $2^{p-1}(2^{p} - 1)$ ( $p$ and $2^{p} - 1$ are prime numbers ) My attempt $\phi(n)$ = ($2^{p - 1} - 2$)($2^{p} - 2$) So, we need to prove that. $2^{p - 1}$($...
Oppenheimer's user avatar
0 votes
1 answer
60 views

Let $x$ and $y$ be non-negative integers such that $N =2^6 + 2^x+2^{3y}$ is a perfect square and $N\leq10,000$. Find the maximum value of $x+y$.

Let x and y be non-negative integers such that $2^{6}+2^{x}+2^{3y}$ is a perfect square, and the expression should be less than 10,000. Find the maximum value of $x+y$ $2^{6}+2^{x}+2^{3y} <= 10,...
Jacob Phan's user avatar
0 votes
2 answers
45 views

Prove that the only natural value of $g$ that makes $8g+4p^2-4p+1$ ($p\in\mathbb{N}$) a perfect square is $g=p$

This problem came up while I was working on a larger problem and I've looked at a few different ways of solving it, the main approach being mathematical induction; however, I've been unable to prove ...
b_rop's user avatar
  • 68
1 vote
2 answers
47 views

What is the reason for the ratios of square units not being the same as the ratios of units [closed]

When you have a square with side length 1 yard you get an area of 1 yd$^2$. When you convert the units of this square to feet you get a square with side length 3 and therefore an area of 9 square feet....
Sam's user avatar
  • 23
3 votes
3 answers
87 views

Prove that number $(4n-1)(2n-1)(2n+1)(4n+1)$ is not a perfect square.

the problem Prove that whatever $n$, a non-zero natural number, the number $(4n-1)(2n-1)(2n+1)(4n+1)$ is not a perfect square. my idea I tried working with modular arithmetic in congruences modulo 4, ...
IONELA BUCIU's user avatar
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0 votes
1 answer
68 views

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
74 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
90 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
  • 1,136
0 votes
0 answers
29 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
75 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
51 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
116 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
  • 1,136
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
  • 1,136
2 votes
2 answers
86 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
83 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
104 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
  • 1,136
2 votes
3 answers
179 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
210 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
  • 1,136
2 votes
1 answer
91 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
  • 1,136
5 votes
3 answers
185 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
110 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
67 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
354 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
124 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
183 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
  • 1,136
3 votes
1 answer
62 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
  • 1,136
0 votes
4 answers
89 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
  • 1,136
-1 votes
1 answer
37 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
  • 1,136
1 vote
1 answer
65 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
136 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
66 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
92 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
  • 1,136
1 vote
1 answer
172 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
  • 1,136
1 vote
2 answers
85 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
  • 1,136
0 votes
0 answers
46 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
  • 1,136
1 vote
2 answers
95 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
  • 1,136
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
201 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
105 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
40 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
88 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
  • 1,136
0 votes
2 answers
181 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
  • 8,415
0 votes
2 answers
97 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
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