# 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 ...
1 vote
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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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### 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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### 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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### 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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### 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 ...
1 vote
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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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### 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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### 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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