# 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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### Prove that $3(p^2)+61$ is a perfect square only for $p = 1$

It is too be proved that $3(p^2)+61$ is a perfect square only for $$p=1 ; p=N$$ The question arises from a problem in arithmatic progression with a certain soft constraint of natural numbers. At the ...
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### $m$ and $n$ are odd numbers such that $|m^2 - n^2 + 1| \mid (n^2 - 1)$. Prove that $|m^2 - n^2 + 1|$ is a square number.

$m$ and $n$ are odd numbers such that $$\large |m^2 - n^2 + 1| \mid (n^2 - 1)$$. Prove that $|m^2 - n^2 + 1|$ is a square number. I have provided the solution below, but I am uncertain about the ...
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### a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
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### Number theory: natural sqrt of 2nd degree polynomial above the naturals.

I am trying to find solutions to $y =\sqrt{ax^2+bx+c}, x \land y \in \mathbb{N}$. I've looked at Sqrt of polynomial, How to find integer X that gives integer y but the solution there assumes $a=1$, ...
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### Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$?

Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$? Let, $2a-b=k^2, a-2b=p^2, a+b=q^2$. $k^2=p^2+q^2$ after ...
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### Why is the sqrt of a matrix different to ^0.5

If H = [4 4; 4 4] Why is sqrt(H)=[2 2; 2 2] And, (H)^0.5=[1.4142 1.4142; 1.4142 1.4142]
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### If a nine digit number is formed by the nine non zero digits with units digit $5$, prove that it must not be a perfect square. [duplicate]

If a nine digit number is formed by the nine non zero digits with units digit $5$, prove that it must not be a perfect square. Greetings, I was continuing on with my number theory, and a question ...
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### How to prove that the difference of squares of two natural numbers can't be a perfect square if their sum is a perfect square? [duplicate]

I came across the following problem and wrote up the following proof a few years ago - what do you think? It is required to demonstrate the truth of Statement S1: "If $a^2 + b^2$ is a perfect square, ...
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### $a, b$ are the integer part and decimal fraction of $\sqrt7$ find integer part of $\frac{a}{b}$

$a, b$ are the integer part and decimal fraction of $\sqrt7$ find integer part of $\frac{a}{b}$ using calculator : $\sqrt 7$ = 2.645 $\frac{a}{b} = \frac{2}{0.6} = \frac{20}{6} = 3.333$ integer part ...
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### Any positive integer can be written as sum / difference of consecutive squares

How should one go about proving that $x \in \mathbb{N}$ can be written (with the right combination of signs) as $\pm 1^2 \pm 2^2 \pm \ldots \pm n^2$ for any $x : x, n \in \mathbb N$? I have tried for ...
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### Simple, algebraic issues.

I'm doing some geometry exercises. The first thing I did was inscribe a circle within a square (gave them the same diameter) and found the ratio between the areas to be Circle/Square=$\pi$/4. The ...
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### Squares in different bases

In base 5, 121 is 25+10+1=36=6*6 In base 6 121 is 36+12+1=49=7*7 In base 7 121 is 49+14+1=64=8*8 In base 5 144 is 25+20+4=49=7*7 In base 6 144 is 36+24+4=64=8*8 Can someone please explain why ...
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### Given that a is an integer, find possible $a$'s so that $2^{2a+1}+2^a+1$ is a perfect square [duplicate]

I found this question in an old Olympiad from my country, but the solution I found only provides a brute force approach. The solution said $0$ and $4$, and when it reaches the point it tests $8$, ...
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### Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$. Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$. Is that ...
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### Number of ways of proving that a number given in algebraic form is a perfect square or is not.

I've listed a few ways by which it can be proved. Please correct me if any of these are wrong. Are there other possible ways which I'm missing out? If an even number is a perfect square, it must be ...
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### When are numbers of the form $m^2+9k^2\pm k$ perfect squares?

In pursuing a question of personal interest, I need to characterize solutions to the equation $m^2+9k^2\pm k=c^2$; in other words, I want to know when numbers of the form $m^2+9k^2\pm k$ are perfect ...
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### Finding a perfect square within an interval

I have a function $f(x) = 4x^2 + 4ax + b$ where $a$ and $b$ are both even numbers. I also have an interval within which there are only two integers $x$ that will result in a perfect square. $a$, $b$, ...
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### Diophantine equation $dx^2-y^2=d-1$ for non-square $d$

For $d=2$ or $3$ the continuous fractions for $\sqrt 2$ and $\sqrt 3$ help (at least partially), but for $d\ge 5$, this doesn't seem to work. Is there any known solution or technique to solve this ...
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### How to solve Shonk Sequences?

A Shonk sequence is a sequence of positive integers in which each term after the first is greater than the previous term, and the product of all the terms is a perfect square For example: 2, 6, 27 ...
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### Why does this pattern in powers happen? [duplicate]

Hopefully this not to frivolous to ask, I have wondered it for years. There is a strange pattern in the differences between various power. For example if you square the numbers from one to ten the ...
Find the value(s) of positive integer $n$ such that $n² + 19n + 48$ is a perfect square. I factorised it to $(n+3)(n+16)$, but that gives negative integer answers $-3$ and $-16$. What do I do?