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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3
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0answers
60 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 any natural number $n$. Does it follow that $x=y$? From this question we know the ...
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1answer
37 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
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1answer
11 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 $2x+1$ ...
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2answers
45 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
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2answers
59 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 ...
1
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2answers
87 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 ...
0
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2answers
62 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, ...
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2answers
39 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......
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5answers
106 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 ...
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0answers
42 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 ...
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}...
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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 ...
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0answers
52 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
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1answer
77 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 ...
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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 ...
1
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3answers
51 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, ...
3
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4answers
44 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 $98$ ...
2
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1answer
47 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 ...
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4answers
44 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 ...
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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 ...
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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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3answers
32 views

If $0<r^2<p\in\mathbb{Q}$, there exists positive integer $k$ such that $(r+1/k)^2<p$.

Let $r,p\in\mathbb{Q}$ such that $0<r$ and $0<p$. Assume $r^2<p$. Then, there exists $k\in\mathbb{N}\setminus\{0\}$ for which $$\left(r+\frac{1}{k}\right)^2<p.$$ I am stuck on this ...
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4answers
72 views

Proof to find at least 2005 ordered pairs (x, y)

Prove that there exists a positive integer $N$ such that there are at least 2005 ordered pairs $(x,y),$ of non-negative integers $x$ and $y$, satisfying $x^2 + y^2 = N.$ Not sure how to get going here....
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1answer
40 views

Proving an expression is not a perfect square

Let a, b and c be 3 odd, distinct prime numbers. I have to prove that the product $abc\frac{a+b}2\frac{a+c}2\frac{b+c}2$ cannot be a perfect square. Since a, b and c are prime, we have that $\frac{a+b}...
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1answer
37 views

Prove by contradiction integer ends with 325 [closed]

Prove that a positive integer that ends in 325 can’t be the square of an integer. I'm not sure how to even approach this, I know that $325 = 5^2 \cdot 13$ but that hasn't led me anywhere. Thank you ...
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1answer
49 views

How to check if a number can be represented as difference of a cube and sqaure?

How to check if a number can be represented as difference of a cube and square ? For eg. $18 = 27 - 9$. Hence $18$ can be represented as difference of a cube and square.
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5answers
613 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 ...
1
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4answers
68 views

Why $\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$ is equal to 1?

$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$ By maths calculator it results 1. I calculate and results $\sqrt{-\frac{1}{2}}$. $\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$ $\sqrt{\...
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2answers
70 views

Diophantine Equation with Square Root

I want to resolve the diophantine equation: $\sqrt{x^2+5x+12} ≡ x-2\pmod 5$ I have thought 2 ways: 1. $(\sqrt{x^2+5x+12})^2 ≡ (x-2)^2\pmod 5$ $ x^2+5x+12 ≡ x^2 -4x+4\pmod 5$ $ 9x+8 ≡ 0\pmod 5$ $ 4x+3 ...
4
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1answer
50 views

what loops and points of numbers are possible when you take the alternating sum of the digits of squared?

what loops of numbers are possible when you take the alternating sum of the digits of squared? I've heard about the happy numbers and the sad numbers. if you don't know the happy numbers are numbers ...
6
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4answers
290 views

Square equal to sum of three squares [duplicate]

For which integers $n$ there exists integers $0\le a,b,c < n$ such that $n^2=a^2+b^2+c^2$? I made the following observations: For $n=1$ and $n=0$ those integers doesn't exist. If $n$ is a power ...
1
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2answers
38 views

How to make a perfect square from a number given in a surd form $a+b\sqrt{c}$?

Is there a way of checking that a number can be written as a perfect square and hence finding it if the number is given in the surd form? For example, if I expand and simplify $$(1+\sqrt{2})^{2}=3+2\...
4
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0answers
34 views

is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere?

is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere? just to keep track $\frac{a}{b}$ is the simplest form of $x$ when $x$ is rational. $\frac{c}{d}$ is the ...
2
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3answers
82 views

Prove that $b^2-4ac$ can not be a perfect square

Given $a$,$b$,$c$ are odd integers Prove that $b^2-4ac$ can not be a perfect square. My try:Let $a=2k_1+1,b=2n+1,c=2k_2+1;n,k_1,k_2 \in I$ $b^2-4ac=(2n+1)^2-4(2k_1+1)(2k_2+1)$ $\implies b^2-4ac=4n^2+...
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4answers
349 views

Find all positive integers $n$ for which $1372n^4 - 3 $ is an odd perfect square.

Find all positive integers $n$ for which $1372\,n^4 - 3$ is an odd perfect square. I tried $\bmod ,4,5,7$ and failed. Next, I used Vieta’s Theorem and failed again. Any hints, please. Thank you very ...
2
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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
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0answers
57 views

Proof of the remarkable formula for the n-th non-square?

The OEIS's A000037 entry makes the remarkable claim that every non-square number is given by the sequence $$a(n) = n + \Big\lfloor \frac 12 + \sqrt n\Big\rfloor$$ After looking through the entry, I ...
0
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3answers
81 views

What is the proof that $(a+b)^2 >a^2 + b^2$? [closed]

I would like to know if there is a theorem that proves that $$(a+b)^2>a^2+ b^2$$ where $ab>0$ I am also wondering whether there is a name associated with this inequality.
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0answers
36 views

Expectation of (non-linear) transformed random variables

If I have a two correlated normal distributed random variables, i.e. $A \sim N(m_a,\sigma_a^2)$ and $B \sim N(0,1)$, with correlation coefficient $ Corr(A,B) =\rho$, then I want to define a random ...
2
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1answer
75 views

Is this probabilistic proof for Brocard's Conjecture flawed?

Brocard's conjecture is the assertion that there are at least four primes between consecutive prime squares, for primes greater than 2. In notation, $\pi(p_{n+1}^{2})-\pi(p_{n}^{2})\ge4$ for $n\gt1$ (...
0
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0answers
47 views

How to prove it is a perfect-square. [duplicate]

Let $ x $ and $ y $ be two positive integers. Prove that $$(xy+1) | (x^2+y^2)\;\implies \frac{x^2+y^2}{xy+1}\;\text{ is a perfect square}$$ I assumed there exists $ k\in \Bbb N $ such that $$x^2+y^2-k(...
2
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3answers
68 views

Find a general method to find particular solutions where the sum of the squares of two consecutive integers is equal to the square of another integer

Question: If the sum of the squares of two consecutive integers is equal to the square of another integer, then find a general method to find particular solutions. E.g., $27304196^2+27304197^2=...
0
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1answer
22 views

$A\in \mathbb{F}_q$, then there exists $n\in \mathbb{Z}_{\geq 0}$ such that $A \in \mathbb{F}_{q^n}$ a perfect square.

I was wondering whether the statement in my title holds. I think it does, but I am not sure. I have managed to prove it for the case $A = -3$, but not in the general case. Any ideas or tips?
0
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1answer
55 views

Square root of 1 modulo N

Given a positive integer N, how do we compute $card(A)$ where $A = \{x\in\mathbb{Z}, 0 < x < N \mid x^{2}\equiv1\pmod N\}$, when the prime factorization of N is known. In other words, how many ...
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2answers
43 views

A simple estimation/approximation of square root

Someone works in a kitchen cabinetry workshop and his calculator/phone doesn't have a square root function. He intends to use Pythagorean theorem to calculate right triangle sides for cabinet doors ...
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0answers
53 views

Wondering if anyone knows the rule name for this square observation?

What is the name or rule for this observation on squares that I've explained below? If you're at a number, say 400 (m^2) which is a square of 20 (m), and you want to find the square of 25, that is 5 (...
0
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1answer
50 views

$(N^2 - 1)/(N - 1) = N+1$ and its relation to sorting square matrices.

I'm working on a problem that involves sorting the members of square matrices into groups. Here are the sorting rules: Group size equals the square root of the number of members in the matrix Every ...
2
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0answers
94 views

If $n=N^2m$ for squarefree $m$, then $n$ is the sum of two squares if $m$ has no prime factor of the form $4k+3$

My question is ; Let $n$ be $n=N^{2} m$ , where m is a squarefree integer. Then $n$ can be written that as a sum of two integer squares, if $m$ contains no prime factor of the form $4k+3$. I have a ...
0
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1answer
55 views

Lower Bound for the square root of the sum of squares

I am looking for a good lower bound for the square root of the sum of squares: Let's say we have some known parameters : $x_i > 0$ where i $\in [1,...,n]$ I am looking for a good lower bound of ...
3
votes
2answers
86 views

$x^4-4=y^2+z^2$ prove that it has no integer solution

$x^4-4=y^2+z^2$ prove that it has no integer solution I tried to check mod$4$ , mod $3$ ... It doesn't give anything. I want to solve this problem by supposing that I'm finding the smallest solution ...

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