# 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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### $\lfloor x^n\rfloor\lfloor y^n\rfloor$ 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 any natural number $n$. Does it follow that $x=y$? From this question we know the ...
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### 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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### 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 $2x+1$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### Finding the magic number as following

Let $s$ and $t$ be distinct positive integers with $s+t$ and $s-t$ are a square numbers. A pair $(s,t)$ called magic if there is exist positive integer $u$, such that $12s^2 + t^2 = 4t^2u^3$. Does it ...
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### Prove by contradiction integer ends with 325 [closed]

Prove that a positive integer that ends in 325 can’t be the square of an integer. I'm not sure how to even approach this, I know that $325 = 5^2 \cdot 13$ but that hasn't led me anywhere. Thank you ...
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### How to check if a number can be represented as difference of a cube and sqaure?

How to check if a number can be represented as difference of a cube and square ? For eg. $18 = 27 - 9$. Hence $18$ can be represented as difference of a cube and square.
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### Is it possible that $2^{2A}+2^{2B}$ is a square number?

Let A and B be two positive integers greater than $0$. Is it possible that $2^{2A}+2^{2B}$ is a square number? I am having trouble with this exercise because I get the feeling the answer is no, but I ...
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### what loops and points of numbers are possible when you take the alternating sum of the digits of squared?

what loops of numbers are possible when you take the alternating sum of the digits of squared? I've heard about the happy numbers and the sad numbers. if you don't know the happy numbers are numbers ...
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### Square equal to sum of three squares [duplicate]

For which integers $n$ there exists integers $0\le a,b,c < n$ such that $n^2=a^2+b^2+c^2$? I made the following observations: For $n=1$ and $n=0$ those integers doesn't exist. If $n$ is a power ...