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Questions tagged [square-numbers]

A non-negative integer $n$ is a square number if $n = k^2$ for some integer $k$.

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Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way

Is there a formula that says if a number is the sum of 2 distinct nonzero squares ? This is the sequence I'm trying to emulate: 5, 10, 13, 17, etc.. http://oeis.org/A004431 (Invalid it contains 85, ...
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1answer
48 views

Is this sufficient to prove $\sqrt{n}$ is irrational if $n$ is not a perfect square

If $n$ is a natural number then $n$ is a unique product of primes to integer powers If $n$ is a perfect square then its prime factors will all be to even powers hence when taking the square root the ...
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1answer
43 views

Defining square function in $\mathbb N$ with only $+$ and the predicate “is a square”

I've got some problems with the following: Let $C(x)$ be the unary predicate "is a square" (so, for example, $C(4)$, $C(9)$ are true). I want to prove that the function $q(x)=x^2$ is definable in $(\...
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Square root of high number - method

Consider the number $324$, the square root of which is $18$. You first look at the $2$ last digits $24$. You know that your square root has to end with the unit $2$ or $8$ because of the $4$ at the ...
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2answers
37 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 ...
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2answers
31 views

How to find largest square from given sticks of n length?

We have n number of sticks and each stick of length 2cm , how to form the largest possible square from the sticks without breaking sticks, find area of largest square? Please give me some clue For ...
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70 views

When can we factor $N$ efficiently with a representation $N^2=a^2+b^2$?

Here : Can the sum of two squares be used to factor large numbers? an idea to factor a large number $N=a^2+b^2$ is shown. Under which conditions does a representation $$N^2=u^2+v^2$$ exist , such ...
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If a prime $p$ divides $n^2$ then it also divides $n$ - Is this proof correct?

I'm learning Real Analysis by myself and I wanted to prove that if a prime $p$ divides $n^2$ where $n$ is an integer, then $p$ divides $n$ itself. I saw that proving this is the same as saying that ...
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22 views

Why does square based stacking game always reach the same number of maximum stackings?

The square stacking games works like this: theres a number NP which is the number of pegs on a board. On those pegs balls with numbers can be placed. But they can only be placed on a peg if they are ...
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1answer
87 views

Arrange $n$ tiles in a rectangle pattern

Is there a way to find how to arrange $n$ tiles in order to form a rectangle that is closest to a square? As I result I am looking for the dimension of that rectangle if it exists. For example, here ...
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1answer
52 views

The number of decompositions of $2n-1$ into a difference of two squares?

Is there a exact formula for number of decompositions of $2n-1$ into a difference of two squares? Examples: ...
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2answers
283 views

Finding elements such that none add to a perfect square

Bob asks us to find an infinite set $S$ of positive integers such that the sum of any finite number of distinct elements of S is not a perfect square. Can Bob's request be fulfilled? I can find some ...
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11answers
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Can exact square roots not be found?

I'm brushing up on some higher level maths for a programming project. I was going along and then I realized that I have absolutely no idea how square roots can be computed. I mean sure, I've ...
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3answers
107 views

Divisor of $x^2+x+1$ can be square number?

$$1^2+1+1=3$$ $$2^2+2+1=7$$ $$8^2+8+1=73$$ $$10^2+10+1=111=3\cdot37$$ There is no divisor which is square number. Is it just coincidence? Or can be proved? *I'm not english user, so my grammer might ...
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1answer
65 views

Show that the term $xy+1$ is a perfect square.

Let $F_k$ denote the $k$th Fermat number $2^{2^k}+1$. If $$A=\{F_{2n}, \ F_{2n+2}, \ F_{2n+4}, \ 4F_{2n+1}F_{2n+2}F_{2n+3}\},$$ then I want to show that for $x,y\in A$ with $x\neq y$ the term $xy+1$ ...
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5answers
117 views

How do you find $x$ such as $y \leq \sqrt{2x-x^2}$?

I am unable to isolate the variable $x$ of this inequality $y \leq \sqrt{2x-x^2}$ ( where $0 \leq y \leq 1 $) Is it correct doing this: $y^2 \leq 2x-x^2$? I found that $y^2 \leq x \leq 2-y^2$ and $...
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1answer
30 views

Can a sum of $n$ consecutive perfect squares be written as a sum of $n-1$ different perfect squares?

So, can $\sum_{i=1}^n i^2$ be written as a sum of $n-1$ different perfect squares? Surely if we are looking at this problem with small numbers, the answer is both yes and no. If we take $n$ to be 3, ...
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2answers
35 views

Algebra - Square of natural numbers [duplicate]

$a$ and $b$ are natural numbers, their product $a\times b$ is full/complete sqare, prove that then $a$ and $b$ are full squares. $\gcd (a,b)=1$ . Natural number is full square if you can write it in ...
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Quick! Prove that $3^{15}+37$ is a square.

CONTEXT: I have been studying the odd powers of $3$ and trying to determine when they are "close to" square numbers; more specifically, I have conjectured that there exist finitely many solutions $m,n$...
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4answers
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When I complete the square on $3x^2 - 12x + 14$ I get an imaginary number, where have I gone wrong?

I have a question in my excersise book: By completing the square show that the expression $3x^2 - 12x + 14$ is positive for all $x$ My approach was to complete the square and rearrange to make $x$ ...
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42 views

Find all pairs of numbers, which meet the assumption.

I have to find all pairs of numbers $ c, d \in \mathbb{R}^2 $, which meet the assumption. The assumption is: $ S_n = (c-5d)n^2+ncd^2-2c^2-2d^2+c+d $ is the sum of the first n words of some ...
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Formula to compute partial sums of the Basel problem?

Question What is the taylor expansion of $\Gamma(n+1)$ of the first $4$ terms? Are the results below correct? I thought of an interesting way to calculate the following quantity: $$ \prod_{(s-1)^...
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4answers
49 views

Is a number written in the square root/fraction form called a non-integer even if it can be simplified to an integer [closed]

This is very simple question, but I cannot get the ansewer from the internet. Is a number written in the square root/fraction form called a non-integer even if it can be simplified to an integer. ...
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4answers
43 views

Diophantine equation: solving $a^2+4n=b^2$

I found myself working with diophantine equations but I have no experience at all with them. Given an integer $n$, can I find two integers, $a$ and $b$, such that $$a^2+4n=b^2$$ How would you guys ...
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2answers
40 views

Sum of divisors mod 8

I am wondering if anyone knows if there are any formulas or results about the sums of divisors of an odd perfect square mod 4,8, 12 etc or anything helpful?
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1answer
36 views

Factors of Square Numbers

Let $n$ and $k$ be positive integers. Prove that if $n+k$ is a factor of $n^2$ then $k > \sqrt{n}$. I do not really know how to approach this. I tried letting $(n+k)(n-m) = n^2$ for some positive ...
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1answer
30 views

Consecutive Square Numbers divided by Consecutive Odd Numbers

Why do we get this pattern (note what has been highlighted or italicized) when you divide consecutive square numbers with consecutive odd numbers ? 4 ÷ 3 = 1 R 1 9 ÷ 5 = 1 R 4 16 ÷ 7 = 2 R 2 25 ÷ ...
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Analogies and relations between evenness and squareness in arbitrary rings

Being even and being a square in arbitrary rings $R$ are analogous by definition: $p \in R$ is even if there is an $a \in R$ with $p = a + a =: 2a$. $p \in R$ is a square if there is an $b \in ...
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1answer
68 views

What are perfect squares in general rings?

The definition of a square element can be made for arbitrary rings $R$: $p\in R$ is a square if there is an $a\in R$ with $p = aa$. The definition of a perfect square element for arbitary rings &...
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1answer
51 views

Find all natural numbers $n$ such that $2n+1$ and $3n+1$ are square numbers and $2n+9$ is a prime.

Find all natural numbers $n$ such that $2n+1$ and $3n+1$ are square numbers and $2n+9$ is a prime. I can prove: $n$ divide by $8$ leaves $0$; $n$ divide by $5$ leaves $0$ So $n$ divide by $40$ ...
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1answer
36 views

Primes of the form $p=2a^2-1$ satisfying $p^2=2b^2-1$.

Are there any prime numbers $p$ for which there exist integers $a$ and $b$ such that $$p=2a^2-1\qquad\text{ and }\qquad p^2=2b^2-1,$$ other than $p=7$? The fact that $p^2=2b^2-1$ implies that $$(p+b\...
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Quick algorithm to find the first $n$ integers that can be expressed as a summation and as a perfect square… how does it work?

Edit: so it turns out these are called square triangular numbers I was helping my sister out with her APCS homework and one of the questions was to find the first $n$ "magic squares" where a magic ...
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1answer
38 views

Three Digit Square w/ Numbers 1-9

Using a computer/program, can anyone figure out: The square of 567 is 321,489. These two numbers contain each of the digits from 1 to 9 exactly once between them. What other three-digit number and ...
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Three-Digit Square Containing Numbers 1-9

Question: The square of 567 is 321,489. These two numbers contain each of the digits from 1 to 9 exactly once between them. What other three-digit number and its square have this property? I know ...
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2answers
59 views

Does a factorial always differ by a square from a square?

Something I have noticed: $$ 4!+1=5^2\\ 5!+1=11^2\\ 6!+3^2=27^2\\ 7!+1=71^2\\ 8!+9^2=201^2 $$ And you can go on. What is going on?
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2answers
38 views

Perfect square or cube

There is series like N= 3! + 4! +.....+ 64!. It is asked whether it is perfect square or cube . How to identify Whether N is perfect square or cube for any big factorial or for sum of factorial ?
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Perfectly Repeating Square Grid on a Hexagonal Tile Base

I am attempting to discover what the sizing ratios of squares and hexagons are. What I want to do with this information is determine (if possible) what size squares in a grid I should place over a ...
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33 views

Perfect squares with algebra

Hi this ' True or False' question is in the 'GCSE Mathematics for AQA Higher Student Book' that we use at school $30\frac{1}{4}$ must be added to $x^2 - 11x$ to make a perfect square. I am sure ...
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3answers
31 views

Square root of two positive integers less than or equal to the sum of both integers direct proof

Please help with this problem. If x and y positive integers, show: $$2\sqrt{xy} \le x + y $$
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3answers
172 views

Find all the positive integers k for which $7 \times 2^k+1$ is a perfect square

Find all the positive integers $k$ for which $7 \times 2^k+1$ is a perfect square. The only value of $k$ I can find is $5$. I am not sure how to find every single one or the proof, I simply used ...
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History of square root of a matrix

Im just curious, who invented the so called square root of a matrix? When it was started ? I wonder why there is no any background of it in Wikipedia ? Is it because finding a square root was just ...
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Prove that an odd square cannot be a pseudoprime with both base 2 and base 3

Background: The Baillie PSW primality test 1 tests if the number is a square before the Selfridge parameter selection. The Mathematica implementation of PrimeQ does ...
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2answers
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Find values for which the given expression is a perfect square

Find all prime numbers $p$ such that $38 p+23$ is a perfect square. $p$ can be $ 7, 11, 79$ etc. I think there would be infinitely many primes. Is there any method to determine all the solutions? ...
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2answers
34 views

Avoidance of patterns $\underbrace{0… 0 …0}_{k^{*}\text{ times}}1$ of last digits in squares

I would like to propose a following conjecture, which I believe is true, but do not have an idea on how to approach to a possible soulution. If we take a set of natural squares $1,4,9,16,25,...$ then ...
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4answers
201 views

When is $ 999\cdots$ a perfect square?

I'm interesting to look more about property of this number $ 999\cdots$ , for even digits which form that number it's clear that's not a perfect square for example :$ 99=10^2-1,9999=10^4-1,\cdots$ , ...
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1answer
68 views

Can an integer of a particular form be a perfect square?

Can an integer of the form $27 + 72 n$, where $n \in \mathbb{Z}$, be a perfect square? I just checked the first $100$ squares... would the quad residues be all the numbers relatively prime to $72$? So ...
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0answers
174 views

Are there 3 definitions of rectangular numbers?

I'm a little bit confused about the definition on rectangular numbers, which is also giving me doubts about its function (or idea). So there's the definition of that all rectangular can be found with ...
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1answer
54 views

If $pqr(p+q+r)$ is a square and $p,q,r$ are primes, then what's the maximum value of $p+q+r$?

We have primes $p\leq q\leq r$ such that $pqr(p+q+r)$ is a perfect square. Find $\max(p+q+r)$. The only thing I've noticed is that all three can be the same. Let's say $pqr(p+q+r)=a^2$. Then if $p=q=...
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0answers
53 views

Limits on difference between a perfect square and a perfect cube?

Are there known bounds on these differences? $378661^2-5234^3=17$ is the best I found for small numbers, which suggests to me that there's maybe a lower bound that goes as the cube root of the number ...
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1answer
32 views

Odd amicable pair

This may be a bit of an uneducated guess or question but are there two numbers which are perfect squares, odd and amicable?