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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0answers
48 views

For two odd primes $p<q$, can we deduce positive integers $a,b$ solving $a^2+b^4=pq$ without trial & error (brute force)?

Let us fix two primes $p,q$ with $2<p<q$. How can we find positive integers $a,b$ which solve the equation $a^2+b^4=pq$ without brute force? Interestingly there exist sometimes two solutions: $...
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3answers
35 views

Proving a and b are perfect squares if and only if $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ are perfect squares

Let $a$, $b$ be two positive integers. Prove that $a$ and $b$ are both perfect squares if and only if both $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ are perfect squares. I believe the proof is based ...
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55 views

Is there a tight bound on following binomial summations involving squares on arithmetic progressions?

The summations of interest is following: $$\sum_{i=0}^{\lfloor\sqrt n\rfloor}\binom{n}{i^2}$$ $$\sum_{i\in\{a,q+a,2q+a\dots,\lfloor\sqrt n\rfloor\}}\binom{n}{i^2}$$ where $q<n$ and $a\in\{0,1,\dots,...
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2answers
79 views

How to explain these features of the squares from 01-99?

The last two digits of the square of numbers from 1-99 have some interesting features as observed below. Do we have simple explanations for them? Or are there any other patterns we find? Let the ...
0
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1answer
25 views

prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$

prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$. My try: use gaussian integer to claim that we have ufd. the problem, is that k might not be irreducible ...
2
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1answer
49 views

How do I find a number $\frac{a^4}{4}$ to be added to an odd integer $N$ to make it a Perfect Square?

Finding the least number to be added to an integer $N$ to make it a Perfect Square is simple: https://www.geeksforgeeks.org/least-number-to-be-added-to-or-subtracted-from-n-to-make-it-a-perfect-square/...
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2answers
31 views

$p_{n}$ be the probability that $C+D$ is perfect square. Compute $\lim\limits_{n \to \infty}\left(\sqrt{n} p_{n}\right)$

Assume $C$ and $D$ are randomly chosen from $\{1,2, \cdots, n\}$. Let $p_{n}$ be the probability that $C+D$ is perfect square. Compute $$\lim\limits _{n \to \infty}\left(\sqrt{n} p_{n}\right)$$ My ...
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3answers
43 views

How do you prove for all $k$ in the integers, $2-5k$ can never be a perfect square?

Proving this for $k > 0$ is easy and intuitive, but I'm unsure how to proceed with the negative values for $k$. Should I use modular congruence somehow? Or can I prove this more simply? My gut said ...
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2answers
304 views

Is it possible to split the natural numbers into a finite number of sets so that no pair of numbers within a set adds up to a square?

My attempt: $\lbrace6,19,30\rbrace$ is sufficient to show that two sets are impossible. Using a computer program with a brute force method I found that separating the numbers $1$ through $85$ into ...
2
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1answer
87 views

Is there any situation where $x^2 \not= \sqrt{x^4}$

This is new for me so sorry if i am missing something, thanks for any helpful pointers. So I was thinking about this classic equation $E=mc^2$ Then I was, why is there $c^2$. I found it is just ...
3
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1answer
23 views

Can you prove that if $\mid D_n \mid = m$, where $m$ is odd, then $n$ is a perfect square. [duplicate]

$n,m \in \mathbb{N}$ and $D_n$ is the set that contains all divisors of $n$, including $1$ and $n$. My Claim: if $\mid D_n \mid=$ an odd number $\iff n$ is a perfect square. and indeed Using Ordinal ...
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1answer
76 views

If an arithmetic progression contains a perfect square, then it must contain a perfect square strictly less than …

(Question): If an arithmetic progression of positive integers $a, a+d, a+2d, \dots$ contains a perfect square, then it must contain a perfect square strictly less than $a+2d\sqrt{a}+d^2$. I noticed ...
3
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1answer
98 views

$a+n^2$ is the sum of two perfect squares

$a+n^2$ is the sum of two perfect squares for a given positive integer $a$ and all positive integers $n$. Show that $a$ is a perfect square. At first I thought of putting a bound on the difference ...
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1answer
61 views

Showing that a number is a perfect square [closed]

How do I show that A=999...982000...081 is a square number when the oly information i have is that there are just as many 9s as there are 0s (with the number being greater than or equal to 4) ?
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1answer
82 views

Expressions are not perfect square for any prime number p

For what values of $n \in \mathbb{N}$ does the two expressions $n^2 + 4n + 1 - p$ and $2n^2 + n + 2 - p$ ARE NOT perfect squares for any prime number $p$. some progress - Manually I found out one such ...
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2answers
89 views

To find all integers n such that the given expression is a perfect square

The question is as follows Determine all integers n such that $n^{4}-n^{2}+64$ is the square of an integer Here is my approch Let $n^{4}-n^{2}+64=k^{2}$. On multiplying both sides by 4, we get $$4n^...
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1answer
48 views

Legendre 3 Square problem

I got a little help on the way here yesterday but it seems like my question is dead. Problem: If $ n \in \mathbb{N} $ can be represented as $ n = n_1^2 + n_2^2 + n_3^2 ,\quad n_1, n_2, n_3 \in \mathbb{...
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3answers
141 views

Complete square values of quadratic formula

For Which values of $x \in [0,n]$ the polynomial $P(x)=ax^2+bx+c$ $\ $ gives a complete square value. For instance the polynomial $P(x)=3x^2+5x+7$ on the interval $x \in [0,100]$ has only two values ...
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3answers
32 views

How do we know that if we keep going with a step of two in the natural numbers we will land on a perfect square, either odd or even?

I really need to use this for another proof that I have. Been struggling with this. Can I say that it is trivial? It feels very common sense but would I need to specifically explain it if I do some ...
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3answers
51 views

How many positive integers $n$ are there such that $(7n + 1)$ is a perfect square and $(3n + 1) < 2008$?

How many positive integers $n$ are there such that $(7n + 1)$ is a perfect square and $(3n + 1) < 2008$ ? What I Tried: We have $(3n + 1) < 2008$ $\rightarrow n < 669$ . Now, $(7n + 1) = k^...
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1answer
68 views

Is there a perfect square number n, whose Euler Totient value is also a perfect square?

Start with a perfect square, denoted as a positive integer n. The root of this square is k, another positive integer. Thus n = k^2 Let t = totient(n) Is there a way to prove there is no such number? I ...
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2answers
77 views

Puzzle relating squares to 2020 and 2021 and the question when this happens again?

Take the numbers of the current and next year $(2020,2021)$ I noticed that $ 2020 = 2* 1010 $ and if we take the square of theses divisors added by $2021$ we get the squares $45$ and $1011$. Or also $...
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1answer
45 views

Given two natural numbers $a,b$ so that $ab+\left(a+1\right)\left(b+1\right)=2n=fixed,$ find $a\left(a\leq b\right)$ such that $b-a$ at the least.

Problem. Given two natural numbers $a, b$ so that $ab+ \left ( a+ 1 \right )\left ( b+ 1 \right )= 2n= fixed,$ by programmin' find $a\left ( a\leq b \right )$ such that $b- a$ at the least. I haven't ...
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2answers
366 views

For how many natural numbers $a$ is the expression $\sqrt{\frac{a+64}{a-64}}$ also a natural number? [closed]

For how many natural numbers $a$ is the expression $\sqrt{\frac{a+64}{a-64}}$ also a natural number? I cannot seem to do this without using trial and error.
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1answer
40 views

“If x is a difference of squares, prove that 3x is a difference of squares as well.”

I'm banging my head against the wall with this task: Prove that if $x$ is a difference of integer squares, then $3x$ is a difference of integer squares as well. What strategies could I utilise in ...
2
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1answer
60 views

Which one can be a square number?

Which one can be a square number? $1)\overline{ab\cdots18}$ $2)\overline{ab\cdots42}$ $3)\overline{ab\cdots46}$ $4)\overline{ab\cdots89}$ To solve this question I considered $17^2=289$ and last two ...
2
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1answer
80 views

$\frac{b^{2n}+b^{n+1}+3b-5}{b-1}$ is square

Find all $b>5$ so that $x_n = \frac{b^{2n}+b^{n+1}+3b-5}{b-1}$ is square for all sufficiently so large integers n. I think the only value of $b$ is 10. If there is $p \in \mathbb{P}$ (prime), $p \...
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0answers
10 views

Find N Distinct Evenly Spaced Perfect Squares [duplicate]

I understand that it's possible to find 3 distinct evenly spaced square numbers is possible by finding a rational solution to the equation: $x^2+y^2=2r^2$ Where $r$ is a rational number. But I have no ...
4
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1answer
126 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 ...
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2answers
66 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
4
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1answer
567 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 ...
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1answer
62 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 ...
4
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8answers
395 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 ...
2
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3answers
73 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 ...
0
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1answer
82 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
100 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 ...
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1answer
46 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\...
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0answers
55 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 ...
4
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2answers
113 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
60 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...
3
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2answers
99 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
74 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 ...
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3answers
136 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 ...
18
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1answer
522 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 ...
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1answer
58 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
19 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
72 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
71 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
390 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
79 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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