# 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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### Is there a function that can cancel a square of a number, preserving its initial value?

When we have the square function defined by : $y = x²$ This implies $x = \sqrt{y}$ or $x = - \sqrt{y}$, because the function $f(x) = \sqrt{x}$ can have only one image, and does return only a positive ...
1 vote
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### Is it possible for three distinct positive integers to exist such that sum and difference of every two is a perfect square?

I have been struggling with this problem for quite some time now but i haven't been able to find a solution yet. My first approach was to try and make an Euler's Brick type form but that didn't really ...
1 vote
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### If $x>y\ge1$ are integers, with $x$ even and $y$ odd, and $x^4+4x^3y-4xy^3-4y^4$ is a square, must $y \mid x$?

The title says it all: Let $x$ and $y$ be integers, with $x$ even and $y$ odd, such that $x > y \ge 1$ and $$x^4+4x^3y-4xy^3-4y^4 = w^2$$ for some integer $w$. Does this force $y \mid x$? Brute ...
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### How can we show that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares?

I saw a lot of problems that assume this: $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares. I wonder how can we demonstrate it because i saw a lot of people using ...
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### Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.

QUESTION Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers. MY IDEA I decided to write it as: $2x^2+2y^2-4xy+x^2+x-2=0={(x\sqrt{2}+y\sqrt{2})}^2+x^2+x-2$ I was thinking of ...
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### Square Formula I came up with. Criticize me! [closed]

You can apply this to squares from between $10^2$ up until $99^2$ Multiply the ones place of the to-be-squared number by itself to get the ones place of the solution, add the ones place to itself to ...
1 vote
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### Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$

Let $a$, $b$ and $c$ be nonzero natural numbers. Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$. My ideas For those numebrs to ...
1 vote
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### Find the $n^{th}$ number, $p$, where $p$ is a prime number and $x^2-2$ is evenly divisible by $p$, where $x$ is an integer

I am trying to complete a coding challenge where, given a number, $n$, I need to create a program to output the $n^{th}$ prime number, $p$, for which there exists an integer, $x$, such that $x^2 - 2$ ...
1 vote
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### Calculating square roots using perfect squares

I recently discovered a way to quickly calculate perfect squares (that are close to a prior-known perfect square), then extrapolated from that a method to mentally calculate the square root of numbers,...
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### $a^2+a+3b$ and $b^2+b+3a$ both perfect square

Find all posssible $(a,b)$ where $a$ and $b$ natural number (positive integer) such that $a^2+a+3b$ and $b^2+b+3a$ both perfect square so, i try to assume that $a^2+a+3b=(a+m)^2$ with $m$ natural ...
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### Concatenation of square numbers is a square?

Just a curiosity of mine. If I define the $n^{th}$ concatenation number (denoted $Q_n$) to be the the concatenation of digits from the $1^{st}$ square number to the $n^{th}$, can $Q_n$ ever be square ...
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### Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal? [closed]

Can $1!+2!+3!+...n!$(written in base 18) be a perfect square in decimal? I know that $1!+2!+3!+...n!$ is never a perfect square if $n\geq5$, since the last digit of the sum is $3$, but I don't know if ...
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### Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power?

Is $7$ the largest possible value of $n$ for which $(1!+2!+3!+…n!)+16$ is a perfect power? I noticed that $(7!+6!+5!+4!+3!+2!+1!)+16=77^2$ is a perfect power, and I don’t know if that is the largest ...
1 vote
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### Perfect square in the sequence formed by consecutive numbers

I recently saw a question Is there any perfect square in the sequence $12,123,1234,12345,...$? This led to thinking about a new question. Consider sequence https://oeis.org/A057137 that is the ...
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### Is $1^{1!}+2^{2!}+3^{3!}+\dots n^{n!}$ a perfect square?

Is $1^{1!}+2^{2!}+3^{3!}+\dots n^{n!}$ a perfect square? Obviously, $1$ is a perfect square. When $n=2$, the sum is $5$, which is not a perfect square. When $n=3,4$, the sum is $2\pmod{4}$. For $n=5$, ...
Let $S=k*m+p$ be any number where $k,m\in\mathbb{N}$ and $p$ is a prime number. My claim is the following: There exists some nondecreasing function $f:\mathbb{N}\mapsto\mathbb{N}$ with $f(m)\to\infty$ ... 