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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4
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1answer
177 views

is 0 a perfect square

Based on my research, I found that there are many arguments about this statement, the main factor is the true definition of perfect square. Some said they are the squares of the whole numbers, but ...
-1
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2answers
61 views

Find all pairs of primes $p, q$ such that 3$p^2q+16pq^2$ equals to square of natural number [closed]

Find all pairs of prime numbers $p, q$ such that 3$p^2q+16pq^2$ equals to square of natural number My attempt: I've been trying to calculate equation through square root but now I'm stuck, please help
1
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0answers
61 views

Show that $4x^2-yz$ is a perfect square

Here is my problem. $A=xy+yz+zx$, where $x,y,z\in\mathbb{Z}$. It is known that if we add $1$ to $x$, and subtract $2$ from both $y$ and $z$, the value $A$ won't change. Prove that $-A$ is a square of ...
43
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7answers
4k views

Can any number of squares sum to a square?

Suppose $$a^2 = \sum_{i=1}^k b_i^2$$ where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct). Can any positive integer be the value of $k$? The reason I am ...
2
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3answers
183 views

Deciding if a number is a square in $\Bbb Z/n\Bbb Z$

I am looking for a systematic way of deciding if a given number is a square in $\Bbb Z/n\Bbb Z$. E.g. is $89$ a square in $\Bbb Z/n\Bbb Z$ for $n\in \{25,33,49\}$? Brute-forcing it would take too ...
-2
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1answer
56 views

Prove that $\sqrt{n^2 + 1}$ is not an integer [closed]

Prove that $\sqrt{n^2 + 1}$ is not an integer The title sums it all... I have tried to prove it for the past hour but I'm just stuck... Tried using induction and assuming the opposite. The logic is ...
5
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8answers
341 views

Proof: not a perfect square

Let $y$ be an integer. Prove that $$(2y-1)^2 -4$$ is not a perfect square. I Found this question in a discrete math book and tried solving it by dividing the question into two parts: $$y = 2k , y = 2k ...
9
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1answer
338 views

Conjecture : an odd perfect square $n>1$ raised to the $m$-th power is never divisible by the sum of $n$'s divisors

This is a conjucture that I created : Let $\,n = (2k+1)^2 \,\, $with $k\in \mathbb{N}$ and so $n>1$, and let $$\,\,A = \sum_{d \in \mathbb{N}; \ d|n} d.$$ Then $n^m$ is never divisible by $A$ ...
2
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3answers
55 views

Perfect square involving the exponential law

If $n$ is a natural number, and $2^{10} + 2^{13} + 2^n$ is a perfect square, what is the value of $n$? I've attempted to factor out $2^{10}$ and got $2^{10}(1 + 2^3 + 2^{n-10})$. How can I move ...
17
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1answer
474 views

$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
0
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1answer
80 views

What's the problem with $-2=(-8)^{\frac{1}{3}}=(-8)^{\frac{2}{6}}=\sqrt[6]{(-8)^{2}}=2$?

$-2 =(-8)^{\frac{1}{3}} = (-8)^{\frac{2}{6}} = \sqrt[6]{(-8)^{2}}=2$ first glance I want to show for $a^{rs}$ to work, both $a^r$ and $a^s$ need to be valid, but as you can see, both $-8^{\frac{1}{3}}...
4
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1answer
94 views

Is there a mathematical formula for the nearest-square function?

Let $x$ be a positive integer. Is there a mathematical formula for $$f(x)=\text{nearest square to } x \text{ }(\text{in terms of } x)?$$ I tried searching for related questions in MSE and found this ...
8
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5answers
13k views

If the square of a number is even, then the number is even. Is that true for 2?

I'll quickly go over my understanding of it: If a number $n^2$ is even, then $n$ is even. The contrapositive is that if $n$ is not even, then $n^2$ is not even. We represent n as $n=2p+1$. $n^2=4p^2 + ...
1
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0answers
52 views

Show that for $u, v \in \mathbb{Z} $ there are only a finite number of $a, b \in \mathbb{Z} $ such that $(ab)^2-ua-vb$ is a square.

This is a generalization of my question Which positive integers $a$ and $b$ make $(ab)^2-4(a+b) $ a square of an integer? Show that for $u, v \in \mathbb{Z}^+ $ there are only a finite number of $a, b ...
0
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1answer
42 views

Pattern with Square Numbers

I have noticed two patterns with square numbers. $1^2\equiv 1\pmod{10}$ $2^2\equiv 4\pmod{10}$ $3^2\equiv 9\pmod{10}$ $4^2\equiv 6\pmod{10}$ $5^2\equiv 5\pmod{10}$ $6^2\equiv 6\pmod{10}$ $7^2\equiv 9\...
4
votes
2answers
108 views

Which positive integers $a$ and $b$ make $(ab)^2-4(a+b) $ a square of an integer?

Which positive integers $a$ and $b$ make $(ab)^2-4(a+b) $ a square of an integer? I saw this in quora, and found that the only solutions with $a \ge b > 0$ are $(a, b, (ab)^2–4(a+b)) = (5, 1, 1)$ ...
0
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1answer
59 views

Showing if the following can be a perfect square

I'm trying to see if $$\frac{n(n^2+1) }{2}$$ can be a perfect square for $n$ a positive integer, but I have no idea how to...
1
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2answers
299 views

Find all natural numbers $n$ for which the equation $x(x+n)=y^2$ does not have any solutions over the positive integers

I tried rearranging it and factoring the sum of squares, so that I get $$xn=(y-x)(y+x)$$ But at this point I have just no clue how to continue. I tried to manipulate the fact that $n$ divides right ...
4
votes
2answers
86 views

Smallest $k$ Such that $13 + 4 \cdot k \cdot p^2$ is a Perfect Odd Square

Given a prime number $p$, I am looking to find the smallest positive integer $k$ such that the following equation $$13 + 4 \cdot k \cdot p^2$$ produces a perfect odd square. All variables are integers....
0
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1answer
71 views

Finding a positive integer that can't be expressed in a certain form

I attended a math speech and the speaker left the following question as an exercise: Which positive integer cannot be expressed in the form $$x^2+2y^2+5z^2+5w^2?$$ I've trying to solve it but I ...
2
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2answers
2k views

Square root of zero

I'm old 35 but starting just now with maths, so sorry if I ask non complex questions. 0 is the only number that just has one square root. Is the explanations for this simply that 0 in arithmetic does ...
1
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3answers
113 views

Show that $3n^4+3n^2+1$ is never a perfect square [duplicate]

I am looking for a proof for the fact that $3n^4+3n^2+1$ can never be a perfect square for a natural number $n>0$. I know for a fact that the statement must be true as it came up as one of the ...
23
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8answers
5k views

Why are the last two digits of a perfect square never both odd?

Earlier today, I took a test with a question related to the last two digits of perfect squares. I wrote out all of these digits pairs up to $20^2$. I noticed an interesting property, and when I got ...
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1answer
46 views

Calculating Square root of decimal number manually. [duplicate]

https://youtu.be/tRHLEWSUjrQ In general, it will be difficult to compute the square root of a decimal number manually? Examples : 50.73 71.21 156.45
0
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1answer
16 views

How to Factor Out a Binomial From a Perfect Square Trinomial

I understand how to factor a perfect square trinomial, but I am unable to see the steps taken to go from $$2x(2x + 1) + (2x +1)$$ to $$(2x +1)(2x +1)\text.$$ If you were asked to factor out $2x+1$ ...
1
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2answers
65 views

Proof of observations on natural numbers being expressed as differences of squares.

Inspired by this Hagon Von Eitzen's answer( https://math.stackexchange.com/a/1591028/789547) I started investigating how I could express natural numbers as differences of squares. Using the method ...
2
votes
2answers
60 views

Sum of digits of square number raised to itself

From testing a few different square numbers, it seems to be the case that when raising a square number to the power of itself, the sum of the digits of the result satisfy the property that the sum of ...
0
votes
2answers
68 views

Prove that there exists no natural number $x$ such that $x^2-6$ is a perfect square

I tried to prove this question using contradiction. I first assumed that there is such a perfect square and then claimed that any perfect square can be expressed in $n^2$, where $n$ is an integer, ...
0
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2answers
48 views

Doubt about whether repunits are square [duplicate]

I have doubts in this exercise. Can a number $A= 111...11$, ($1000$ times the number $1$), be a square? Can a number $B= 111...11$, ($10431$ times the number $1$), be a square? Can a number $C= 111......
4
votes
5answers
108 views

When is $k^4-24k+16$ a perfect square.

When is the equation $k^4-24k+16$ perfect square. (k is an integer.) I got this equation as discriminant while solving an equation. I tried to solve it but couldn't i tried to write it in a form of a ...
0
votes
0answers
45 views

How many ways are there to prove that the product of 8 consecutive positive integers is never a square?

I saw this on quora. Here is my solution. (I know about general theorems (Erdos, Erdos-Strauss) that imply this. I wanted a proof that worked for only this special case (8 integers, square).) All ...
15
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8answers
18k views

IMO 1988, problem 6

In 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. ...
2
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0answers
92 views

How do I prove 8(x)! + (2x-1)² where x is an integer greater than or equal to 3, is never a perfect square.

$8(x)! + (2x-1)^2 = a^2$, so $8(x)! = a^2-(2x-1)^2$, for $x \geq 8$, $64|\text{LHS}$, but we can see 8 doesn't divide both of $(a-2x+1)(a+2x-1)$, so 64|one of them, so both one of them is $0 \pmod {64}...
2
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3answers
62 views

Finding the magic number as following

Let $s$ and $t$ be distinct positive integers with $s+t$ and $s-t$ are a square numbers. A pair $(s,t)$ called magic if there is exist positive integer $u$, such that $12s^2 + t^2 = 4t^2u^3$. Does it ...
1
vote
0answers
56 views

A peculiar conjecture including maximal prime factors

Let $\operatorname{gpf}(n)$ be the greatest primefactor of n. By experimenting I found the following conjecture. Given $m,n\in\mathbb N_{>1}$. Then $m+n=\operatorname{gpf}(m)\cdot\operatorname{gpf}...
4
votes
3answers
2k views

At least one member of a pythagorean triple is even

I am required to prove that if $a$, $b$, and $c$ are integers such that $a^2 + b^2 = c^2$, then at least one of $a$ and $b$ is even. A hint has been provided to use contradiction. I reasoned as ...
3
votes
4answers
46 views

How many values of $x\in\mathbb Z^+,x<99$ are there such that $m,n\in\mathbb Z$ and $m^2-n^2=x$ is possible?

How many values of $x\in\mathbb Z^+,x<99$ are there such that $m,n\in\mathbb Z$ and $m^2-n^2=x$ is possible? So what I'm trying to find here is the number of integers between $1$ and $98$ ...
0
votes
2answers
92 views

2041 distinct natural numbers such that the sum of their squares is a perfect square

Determine if there are 2041 distinct natural numbers such that the sum of their squares is a perfect square. Can anyone please help me to solve this problem?
4
votes
1answer
78 views

In which base $b$ is $(374)_b$ a perfect square?

If you convert this number to base 10, we can obtain the expression $$3b^2+7b+4 = (b+1)(3b+4).$$ Since $\gcd(b+1,3b+4) = 1$, we further conclude that both $b+1$ and $3b+4$ are perfect squares. So the ...
0
votes
3answers
48 views

An expression in rational numbers $x, y,$ and $z$: Why is it a square of a rational number?

Let $\,x,y,z\in\mathbb Q\,$ satisfy $\,xy+yz+zx=1$. Given this, I would like to prove that $$\big(1+x^2\big)\big(1+y^2\big)\big(1+z^2\big)$$ is the square of a rational number $n$. That is, you can ...
13
votes
4answers
5k views

Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number ...
1
vote
3answers
53 views

Generalizing $\,r(n^2) = r(n)^2,\,$ for $\,r(n) := $ reverse the digits of $n$

I'm assuming this theorem was found by someone else before, but I found this relationship between square numbers of 3 digits or less. The theorem is this: If you reverse the digits in a square number, ...
2
votes
1answer
51 views

Establishing infinitely many primes of the form $4k+1$.

Establish that there are infinitely many primes of the form $4k+1$. I was studying primitive roots, and it was recently proven that the odd prime divisors $n^2 +1$ are all of the form of $4k+1$. A ...
0
votes
4answers
45 views

Square Triangle numbers and Pell's equation

So i have just started to study number theory and i was asked this question. Now i tried to search online and i found out pell equation can used to solve this question. Now in an online video i saw ...
7
votes
5answers
621 views

Is it possible that $2^{2A}+2^{2B}$ is a square number?

Let A and B be two positive integers greater than $0$. Is it possible that $2^{2A}+2^{2B}$ is a square number? I am having trouble with this exercise because I get the feeling the answer is no, but I ...
5
votes
3answers
240 views

Find all integers $m,\ n$ such that $m^2+4n$ and $n^2+4m$ are both squares. [closed]

Find all integers $m,\ n$ such that both $m^2+4n$ and $n^2+4m$ are perfect squares. I cannot solve this, except the cases when $m=n$.
1
vote
1answer
105 views

Finding all positive integers $n$ such that $\{n,n+1,n+2,n+3,n+4,n+5\}$ partitions into two subsets such that the products of elements are the same [closed]

Question from 12th IMO: Find all positive integers $n\in\mathbb Z^+$ such that the set $\{n, n + 1, n + 2, n + 3, n + 4, n + 5\}$ can be split into two disjoint subsets such that the products of ...
2
votes
1answer
103 views

How many squares in a three-dimensional $n \times n \times n$ cartesian grid?

This brings the classical question to three dimensions. Given a three-dimensional Cartesian grid of $n \times n \times n$ points (that is $(n-1) \times (n-1) \times (n-1)$ unit cubes), how many ...
2
votes
5answers
53 views

Using the fundamental theorem of arithmetics

I’ve got a problem that I’m not quite sure how to solve. I can see the reasoning behind the problem, but I’m not sure how to apply the theorem. Suppose that $a$, $b$, and $c$ are integers, and $a^...
1
vote
1answer
49 views

Roots of imperfect square as sum of other real numbers

Can square roots of imperfect square such as $ \sqrt{2}$ , $ \sqrt{3}$.....$ \sqrt{n}$ be written as sum of other real numbers or other imperfect square roots which are not linear combinations with ...

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