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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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1answer
29 views

Finding solutions in $\mathbb{Z}_{+}$

Find all the triplets of positive integers $(a, b, c)$ such that: $a^2+b+3=(b^2-c^2)^2$
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3answers
36 views

Square numbers related to square root

Given whole numbers $a$, $b$, $c$ satisfying $\sqrt{a}+\sqrt{b}+\sqrt{c}$ are also whole numbers. Prove that $a$, $b$, $c$ are square numbers.
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1answer
44 views

finding perfect squares solutions for the following case

I was working on a number theory problem and create a equation. I tried research on this, but tbh I don't even know what should I google for... Here's my cases. $$n = \sqrt{N * \frac{1+\sqrt{4k^2+1}}...
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1answer
35 views

Product of pairwise sum is perfect square

For which $n$ can we divide $1,2,\ldots,2n$ into $n$ pairs so that the product of the sum of the $n$ pairs is a perfect square? If $n$ is even, this is possible: match the first number with the last, ...
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1answer
29 views

Number multiplied by itself does not give a square number

The answer to this is probably very simple but while working on a question I was surprised to discover than a number multiplied by itself does not give the same answer as the same number squared (in ...
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2answers
49 views

How many ABC three digit numbers are there such that (A+B)^C has three digits and is a power of 2?

I am about to give an exam soon for joining an algorithms course so they sent me a sample test just so I know what's it gonna like and I couldn't solve this problem. The question is the following: ...
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2answers
60 views

$n$-th term of the series 1 27 125 1000

What will be the nth term of the series 1 27 125 1000 for $n = 1$, it is 1 for $n = 2$, it is 27 for $n = 3$, it is 125 for $n = 4$, it is 1000
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2answers
51 views

nth term of the series 1, 16, 24, 1024

What will be the formula for finding nth term of the series for eq for n = 1 it will be 1 for n = 2 it will be 16 for n = 3 it will be 100 for n = 4 it will be 1024 And am i doing it the ...
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4answers
83 views

Finding the mistake of my fake proof where $\pm-2=-2$ [closed]

I am wondering what has gone wrong in the following: $4=4 \iff \sqrt{4}=\sqrt{4}\iff\pm 2= \sqrt{4}\iff \sqrt{4}=2\;\text{and}\;\sqrt{4}=-2\iff \pm2=2$ (from substituting back in and obviously ...
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4answers
123 views

Find $n^2+58n$, such that it is a square number

I have the following problem. Find all numbers natural n, such that $n^2+58n$ is a square number. My first idea was, $n^2+58n=m^2$ $58n=(m-n)(m+n)$ such that $m-n$ or $m+n$ must be divisible by 29,...
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0answers
19 views

Algorithm to find square numbers with certain distance

Is there an efficient way to calculate for a given $c \in N$ two square numbers $x^2,y^2$ with $x^2-y^2=c$, without being able to factorize $c$? I was just thinking about RSA, where $n=pq$ is given. ...
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1answer
124 views

Determine all integers $i$ such that $(i-29)(i+29)$ is a square number

Determine all integers $i$ such that $$(i-29)(i+29)$$ is a square number. I’ve tried some substitutions but none of them worked... I think that the only solutions are $i=\pm 29$, but I still ...
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2answers
42 views

Does there exist an infinite geometric progression whose terms are all squares

I am aware of the fact that the squares don't contain an infinite arithmetic subsequence, but I was wondering if the squares contain an infinite geometric sequence. In other words, does there exist ...
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3answers
77 views

Why is $(5\sqrt{5p}-3\sqrt{5q})(5\sqrt{5p}+3\sqrt{5q}) \equiv 5(5p-3q)(5p+3q)$?

I was working on the difference of two squares, $125p^2-45q^2$ Writing my answer, $$(5\sqrt{5}p-3\sqrt{5}q)(5\sqrt{5}p+3\sqrt{5}q),$$ onto Pearson, I got a popup that said my answer was equivalent ...
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1answer
55 views

Number theory: square numbers of a given form

I made a proof in an undergraduate number theory class which uses this assumption to make a crucial step in the proof. Could someone tell me if it is correct and maybe explain why or why not? If I ...
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0answers
23 views

A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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1answer
33 views

Find amount of square integers between two points

I was recently trying to solve a programming problem on hackerrank and the problem description was that given two points $a$ and $b$ I was supposed to find amount of square integers in given range. ...
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4answers
72 views

For which integer $n$ is $28 + 101 + 2^n$ a perfect square?

This question For which integer $n$ is $$28 + 101 + 2^n$$ a perfect square. Please also suggest an algorithm to solve similar problems. Thanks Btw, this question has been taken from an Aryabhatta ...
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2answers
79 views

Show that if $a,b,c\in \mathbb{N}$ and ${a^2+b^2+c^2}\over{abc+1}$ is an integer it is the sum of two nonzero squares

I was reading about the fascinating problem in the IMO ($1988 $ #$6$) that asks: Let $a$ and $b$ be positive integers such that $(ab+1) | (a^2+b^2)$. Show that ${a^2+b^2}\over{ab+1}$ is a perfect ...
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4answers
1k views

How To solve This Perfect Square Word Problem

Here's a problem about perfect squares and it's very hard for me. I tried to solve but I got stuck. Last year, the town of Whipple had a population that was a perfect square. Last month, 100 ...
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3answers
119 views

perfect square of summation of odd numbers

This is not homework , and I am old enough to be your father :-).We know the summation of odd numbers results in perfect squares , like $1 + 3 = 4 $, $1+ 3 + 5 = 9$ and so on. My question is , if ...
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2answers
77 views

prove that [p!/(p-4)!] + 1 is a perfect square for all natural p.

one can observe that $[p!/(p-4)!] + 1$ is basically the product of four consecutive integers plus one.Since this is $$ \begin{eqnarray} p(p+1)(p+2)(p+3)+1 & = &(p^2+3p)(p^2+3p+2)+1 \\ & ...
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4answers
63 views

What is the value of $abc0ac$, a six digit perfect square number which is divisible by $5$ and $11$?

$abc0a$c is a six digit perfect square number which is divisible by 5 and 11. Find out the number. Source: Bangladesh Math Olympiad 2016 Junior Category I tried but failed to find any relation ...
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0answers
78 views

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
58 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
47 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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1answer
104 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
80 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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0answers
77 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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2answers
101 views

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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0answers
23 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
97 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 ...
2
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1answer
57 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
291 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
5k views

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
112 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
69 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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4answers
157 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
31 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
38 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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4answers
147 views

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
55 views

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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0answers
45 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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0answers
83 views

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
52 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
45 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
67 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?
2
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1answer
37 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
56 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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0answers
30 views

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 ...