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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Suppose that $p$ and $q$ are both prime numbers where $p > q$. Show that $p - q$ and $p + q$ cannot both be perfect squares.

It's a lot harder when its adding and subtracting because I can't use prime factorization to prove anything. I've gotten a little bit, as all primes (with the exception of $2$) are odd, and odd + odd ...
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If $a^2 + a + 1 = 0$ find $a^3$

$$a^2 + a + 1 = 0$$ $$(a^2 + a+1) (a-1) = 0(a-1)$$ $$a^3 - 1 = 0$$ $$a^3 = 1$$ This is how I had solved the question by using the identity :- $$a^3 - b^3 = (a - b)(a^2 + b^2 + ab)$$ But the roots of ...
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How many positive integers $n$ are there such that $2n$ and $2n^2+1$ are both perfect squares?

How many positive integers $n$ are there such that $2n$ and $2n^2+1$ are both perfect squares? $n=2$ is the only solution I can find. Are there others?
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Is there a way to ensure that a simplified square root is right?

I was reading this article about square root and they simplify $\sqrt{75}$ to $5\sqrt{3}$. Is there a way to ensure that the answer is correct, going from $5\sqrt{3}$ to $\sqrt{75}$? For example, I ...
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When is $4a^2+4b^2+4a+4b+1$ a square? [duplicate]

Is it possible to determine when $$4a^2+4b^2+4a+4b+1 = ((2a+2b)+1)^2-8ab = (2a+1)^2+4(b^2+b)$$ is a perfect square, assuming $a,b \in \mathbb{Z}$ and $b>0$? I've tried writing it in a few different ...
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The explicit monoid law on the differences of two square integers?

It is known that if $S = \{ x^2 : x \in \Bbb{Z}\}$ is the submonoid, then $\Delta S = \{ x^2 - y^2 : x,y\in \Bbb{Z}\}$ or the set of all differences of squares forms the monoid $4\Bbb{Z} \uplus( 2\Bbb{...
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Perfect square equation $12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2$

Well, I have the following function: $$\text{y}\left(x\right):=12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2\tag1$$ Where $\alpha\in\mathbb{N}$. ...
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What is the maximum relative density of squares congruent to m modulo n for chosen m and n? [closed]

Let f(m,n) be the number of values of k between 1 and n such that k^2 is congruent to m modulo n. I call f(m,n)^2/n the relative density of squares modulo n. For m = 1 and n = 24, this relative ...
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Is the polynomial $p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$?

My initial question in the present post is pretty basic: Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$? When $k=...
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Why is $\pm\sqrt{(11y-8)^2} = \pm(11y-8)$? - Analytic Geometry

$3x^2-7xy-6y^2-2x+17y-5=0$ My original goal here was to know whether or not this was a degenerate conic, so I isolated $x$ by applying the Quadratic Formula. $3x^2+(-7y-2)x+(-6y^2+17y-5)=0$ $x = \frac{...
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Follow-up to MSE question 3738458

This is a follow-up inquiry to this MSE question. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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Updating standard deviation without set

Say for instance that I have this set: 16, 76, 48, 44, 4, 2, 94, 87, 10, 22 And I calculate the standard deviation for it: Get the mean (we'll call it "m") For each number: (nr - m)^2 (we'...
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Is $(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$ true for all integers $c>1$, when $n$ is an odd integer?

Let $n$ be an odd integer. Is $$(1+c^2)^n-\lfloor(1+c^2)^{n/2}\rfloor^2<(1+c^2)^{(n+1)/2}$$ true for all integers $c>1$? Notes: $c=1$ has a counterexample $2^{31}-\lfloor2^{31/2}\rfloor^2>2^{...
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Number of values of $n\in \mathbb{Z}$ such that $n^2+n+2$ is a perfect square [duplicate]

My approach: Case $1$: $n$ is positive: Let $x\in\mathbb{Z}$, so $$n^2+n+2=x^2$$ $$\implies n+2=x^2-n^2=(x+n)(x-n)$$ $$\therefore x-n=\frac{n+2}{x+n}$$ $\because x\in \mathbb{Z}$ and $n\in \mathbb{Z}$,...
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2 answers
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How could I prove / disprove that every non-zero integer can be written in the form $p-x^2$ where $p$ is a prime and $x$ is a positive integer?

Question: Can every non-zero integer be written in the following form? $$p-x^2$$ I was thinking about if every non-zero integer could be written in the form $p-x^2$ where $p$ is a prime and $x$ is a ...
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The generalized form of $n$ which satisfies $(n+2)^3+(n+1)^2+(n+0)^1 = m^2$ $(n \in \mathbb{N},m \in \mathbb{N})$

I'm thinking about the generalized form of $n$ which satisfies the following equation. $(n+2)^3+(n+1)^2+(n+0)^1 = m^2$ $(n \in \mathbb{N},m \in \mathbb{N})$ From some experiments, I expect that the ...
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2 votes
3 answers
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Interesting Pattern in Square Numbers and Pattern for Cube Numbers [duplicate]

I am absolutely not the first person to notice this, but I did notice that the difference between any two squares increases by 2, starting at 1(between $0^2$ and $1^2$). I am not the best at ...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
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1 answer
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Partial square root of an integer

I am looking for an efficient algorithm to compute what I'd call the "partial square root" of an integer. In more formal terms: Given a positive integer $k$, find the pair of positive ...
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Is this identity $ (1+6x^3+9x^4)^3+(1-6x^3+3x-9x^4)^3+(1-9x^3-6x^2)^3=9x^2+9x+3$ known with $x$ is an integer?

It is known that, let $x$ arbitrary integer $$(9x^4)^3+(3x-9x^4)^3+(1-9x^3)^3=1\tag{1}$$ discovred by Kurt Mahler in 1936, and $$(1+6x^3)^3+(1-6x^3)^3+(-6x^2)^3=2\tag{2}$$ discovred by A. S. ...
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Proving a lemma on rational points

Suppose we have to find if $aX^2+bY^2+cZ^2=0$ has any rational points or not, where $a,\ b,\ c$ are non-zero integers. Define $bc=\alpha^2s, \ ca=\beta^2t,\ ab = \gamma^2u$, where $s,\ t,\ u$ are ...
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1 answer
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If $n$ is a positive integer such that $8n+1$ is a perfect square, then $2 n$ can't be a perfect square

If $n$ is a positive integer such that $8n+1$ is a perfect square, then (a) $n$ must be odd (b) $n$ cannot be a perfect square (c) $n$ must be a prime number (d) $2 n$ cannot be a perfect square Try ...
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1 answer
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contest problem: number theory, prime factorization, perfect squares

Problem Statement: Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the ...
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5 votes
2 answers
299 views

Does a generalized difference of powers formula exist?

The identity $$ \left(\frac{n+1}{2}\right)^2-\left(\frac{n-1}{2}\right)^2=n $$ can be used to represent any number as difference of two squares. (Note that this formula gives integer values when $n$ ...
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Prove $A=\sqrt{x^{2026}-x^{406}+2017}$ is not an integer

I have some difficulty with this problem prove $A=\sqrt{x^{2026}-x^{406}+2017}$ is not an integer I'vs tried to prove $x^{2026}-x^{406}+{2017}$ is not a square number by congruent (with 2 -> 4 ->...
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Conjecture on ordering the first $p^2$ naturals by prime factor count

Let $\text{bump}(n)$ for $n\in\mathbb N$ be a function that increases the prime index of each prime factor of $n$ (with multiplicity) by $1$. I'll also use the notation $\text{bump}^k(n)$ to signify $\...
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3 votes
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Find primes that satisfy conditions

The problem is as follows: Find all primes $p$ and $q$ such that $p-q$ and $pq-q$ are both perfect squares. I found the solution $(3,2)$ by considering when $q$ is even. I then considered when both $p$...
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How to find the closest segment (which is part of a square) of a point on a circle using (or not?) cos() and sin()?

Let's say I have a circle with 1 point (cx, cy) on this circle (whatever the radius - I want to use cos() and sin() to solve this problem so the radius shouldn't matter). I have the coordinates (x1, ...
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1 answer
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Checking whether a number is a perfect square or not.

I was given to tell whether $945729$ is perfect square or not. I used the concept that No number can be a perfect square unless its digital root is $1$, $4$, $7$, or $9$. Digital root of $945729=9$ ...
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2 votes
1 answer
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Are there infinitely many square numbers with increasing digits? [duplicate]

This is a question that came up while joking around with my friends, but now I am really intrigued by this question. For sake of brevity, let's call square numbers with monotone increasing digits ...
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Whole number solutions to $2n^4+1=m^2$. [duplicate]

What are the whole numbers for which two times the forth power of it plus one is a square? In mathematical notation: $$2n^4+1=m^2;m,n\in\mathbb{Z}$$ My Observations: because of the squares, all ...
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Let $w<x$ be integers with $x^2-w^2$ beeing a square. If no $y$ exists with $y^2-x^2$, $y^2-w^2$ being squares, can we show no $y$ exists for $nw<nx$?

There exist vast pairs $(w,x)$, $w<x$ with $x^2-w^2=\square_1$ beeing a square (also known as Pythagorean Triples). I am trying to distinguish two "classes" of such pairs: Those pairs ...
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Can a perfect square N be composed by only 0 and 1? Where N's prime factor are only 3 and 7.

I would like to know if someone knows how to prove that there are no perfect square composed only by zeros and ones in their decimal representation whose prime factors are only 3 and 7 (so of the form ...
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2 answers
263 views

What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.) A number $P$ is said to be perfect if $\...
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2 votes
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Why do we have these occurrences of primes of the form $4k+1$ in square differences (Mengoli's Six-Square Problem)?

For the present investigation Poetasis gave me the necessary impetus in his answer on my related question here. I analyzed a large dataset of triples $w,x,y$ which fulfill the following diophantine ...
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2 votes
2 answers
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find the $x$ of quadratic equation, such that the squre root of quadratic equation result is integer

given this equation $$S=\sqrt{x^2+1500x-1472}$$ find $x$, such that $x$ is positive integer, and $S$ is positive integer. I have tried to solve this, and I get that $x = 36$, but I get that $x$ by ...
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Let $w<x$ be integers with $x^2-w^2$ beeing a square. Which condition they must satisfy for a $y$ to exist with $y^2-x^2$, $y^2-w^2$ being squares?

Let $(w,x)$ be two integers with $x^2-w^2$ beeing a perfect square. One simple example is $(w,x)=(4,5)$, since $5^2-4^2=9=\square_1=3^2$. In this case we cannot find a third integer $y$ which ...
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2 votes
0 answers
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Find all integer pairs $(a,b)$ such that $a^2+ab+b^2=219^2$, without technology.

Find all integer pairs $(a,b)$ such that $a^2+ab+b^2=219^2$, without technology. My colleague showed me this question. He said it is from the AIME (American Invitational Mathematics Examination). I ...
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Find the prime numbers $p, q$ satisfying perfect square

Find the prime numbers $p, q$ satisfy: $p^2+3pq+q^2+6q+6p-60$ is perfect square My try: $p^2+2pq+q^2+6q+6p+9+pq-69=r^2$ so $pq-69=(r-q-p-3)(r+p+q+3)$ and I stuck here, any hint please. Thank you.
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Perfect numbers and Pell's equation

(Disclaimer: The following is a naive attempt to apply the theory of Pell's equations to perfect numbers. Please bear in mind that this is my first time to try solving such an equation.) Let $p^k$ be ...
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1 vote
2 answers
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How we may express four squares whose difference is each a square in terms of (preferably solid) geometry?

The problem of finding four squares whose difference is each a square is much more exhaustive as I thought. A quest up to $2^{34}$ yields nothing. The largest almost solution found in the range up to $...
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-1 votes
1 answer
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How do I prove that $3|m$ and that $m+1$ and $\frac{1}{3}m$ are also perfect squares? [closed]

Let $m$ be a non-zero natural number such that $\frac{m(m+1)}{3}$ is a perfect square. How do I prove that $3|m$ and that $m+1$ and $\frac{1}{3}m$ are also perfect squares?
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1 answer
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How can we make this a perfect square? [closed]

The question I have is: How can this be transformed into a perfect square? $$a(a+1)(a+2)(a+3)+1$$
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Conjecture: between any two consecutive squares, there are integers matching each of $2p, 3p,$ and $4p$; also, more terms with higher degrees

This is a minor twist on Legendre's conjecture, of course. To restate: I submit that for all $n>1$, every interval $\left(n^2,(n+1)^2\right)$ contains at least one integer matching each form $p, 2p,...
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5 votes
3 answers
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Is the set $\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Phythagorean triple with}\ a<b<c\ \right\}\ $ dense in $\ [0,1]\ ?$

Is the set $\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Pythagorean triple with}\ a<b<c\ \right\}\ $ dense in $\ [0,1]\ $ and how do you show this? It seems likely true based on ...
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1 vote
3 answers
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For a perfect square $n$, can we calculate integers $0<y<l$ up to a limit $l$ such that $n+y^2$ is a perfect square (without bruteforce)?

I want to implement a fast algorithm (avoiding or at least minimizing bruteforce) which for a given square number $n$ calculates a series of positive integers $y$ up to a certain limit $l$ such that $...
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10 votes
1 answer
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Does every strip of positive irrational slope contain a perfect square point?

Does every strip between two parallel lines of positive irrational slope contain a point with perfect square coordinates? Equivalently (I think), are there perfect square points arbitrarily close to ...
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1 vote
1 answer
79 views

For a given integer $n>0$, can we efficiently determine whether an integer $y$ exist such that $n+y^2$ is a perfect square?

I am trying to implement an efficient algorithm which must determine whether for a given integer $n>0$ another integer $y$ exist such that $n+y^2$ is a perfect square. Based on some properties of $...
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1 vote
1 answer
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Is this lemma about perfect squares correct or not?

Given the following equation: $$a\times b=y^2$$ Where a,b and y are integers. One of these two things must be true. Either both a and b are prefect squares or a and b are identical. i) For example 2×2=...
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1 vote
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Difference between increasing integer values of $C=x^2+y^2+z^2$

Given are non-negative integer variables $x$, $y$ and $z$. I am trying to deduce the absolute difference between a certain value of $C=x^2+y^2+z^2$ and the very next smallest increase in $C$ possible. ...
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