# 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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### Prove that there exists a positive integer $N$ such that the equation x^2 + y^2 = N has at least 2005 solutions in non-negative integers $x$ and $y$. [closed]

Hint: Try to construct a bunch of triples that all have the same hypotenuse.
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### How to show that $16n^2 + 16n+1 \neq m^2$ with $n, m \in \mathbb{N}$?

How to show that $16n^2 + 16n+1 \neq m^2$ with $n, m \in \mathbb{N}$ ? I already know that $m$ needs to be an odd number because: $16n^2 + 16n + 1 \equiv 1 \mod{4}$ I can complement the square by: ...
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### Solutions to $k$ when $2^k n^2 + 2^k n + 1$ is never a perfect square.

I need to find possible values of $k$ with $k \in \mathbb{N}$ such that for any $n \in \mathbb{N}$ the equation $2^k n^2 + 2^k n + 1$ will never be a perfect square. So, I thought, maybe for even $k$ ...
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### count number of a solution of an integer equation

How to count all couple of strictly positive integers (u,v) such that the following expression also lends an integer? $$\frac{v}{u}\cdot \left[2v-\sqrt{\left(2v\right)^2-u^2}\right]$$ with \begin{...
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### Show that if n is an even perfect number then n is not the sum of two squares. [closed]

A perfect number is a positive integer that is equal to the sum of its proper divisors, and all perfect numbers are even. A sum of two squares is an integer that is the sum of two squares integers.
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### Insights for proving $(x^2-1) = l(l+1)(l+2)(l+3)$ [closed]

I am looking for some insights on how to prove that for every integer $l$, there exists an integer $x$ such that: $$x^2-1 = l(l+1)(l+2)(l+3).$$ So far, what I did was: $$x^2-1=(x-1)(x+1).$$
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### $100-50\sqrt3$ to be a perfect square

If we want $100-50\sqrt3$ to be a perfect square $(a+b)^2=a^2+2ab+b^2$, most likely it is the case that $-25\sqrt3$ corresponds to $ab$. I can't approach the problem further. Can you give me a hint?
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### Finding all four-digit perfect squares of the form $XXYY$ [closed]

Can you find a four digit number of the form $XXYY$ using only mathematical tools (without a computer) where the first two digits are same ($XX$) and the last two digits are same ($YY$), and the ...
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### Find all 6-digit squares which are the concatenation of three 2-digit squares

I am looking to find perfect squares $\overline{abcdef}$ with the property that $\overline{ab}$, $\overline{cd}$ and $\overline{ef}$ are perfect squares. I ran a quick program to find that the only ...
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### Number Theory Prove a complete square 1,11,111 [closed]

I have to prove that every number in the series 11,111,1111 ... Is not a complete square, Can you give me a clue how to do this?
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### $x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square

Determine 4 distinct $A, B, C, D$ for any rational $x$ such that either $x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square. or, $x^3+Cx+D$ is not a square number but $x^3+Ax+B$ is a ...
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### Proving technique (multiples) [duplicate]

If a and b are integers, if a^2 is a multiple of b^2, will a then become a multiple of b? I feel it's true , 1,4,9,16,25,36,49,64.... I tried many. But I forgot how to formally prove it, how to ...
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### Infinitely many squares of form 50^m - 50^n? [closed]

I wanted to solve this problem. You have to prove that there are infinitely many square numbers of the form $(50^m - 50^n)$ (and no square numbers of the form $(2020^m + 2020^n)$ with $m$ and $n$ ...
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### Square numbers and divisibility

I'm trying to prove the following statement: If $kn+1$ and $(k+1)n+1$ are both perfect squares, then $2k+1$ divides n, where k and n are positive integers. Case $k=1$: Put $n+1=x^2$, $2n+1=y^2$. ...
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### Does there exist $n\in\mathbb{N}$ such that $5^n-2^n$ is a perfect square?

I checked up to $n\leq 100000$ then found no example. So I suspect that there doesn't exist $n\in\mathbb{N}$ such that $5^n-2^n$ is a perfect square. Then I tried to prove by modular arithmetic, but ...
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### gnomes and lights vs. sieve of Eratosthenes

I was riddled a riddle the other day - a series of gnomes flips a series of light switches. The first gnome flips every switch, the second gnome flips every second switch, the third gnome flips every ...
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### Prove that $A^2=a_1^2+a_2^2-a_3^2-a_4^2$ for all integers $A$.

Prove that: For every $A\in\mathbb Z$, there exist infinitely many $\{a_1,a_2,a_3,a_4\}\subset\mathbb Z$ given $a_m\neq a_n$ such that $$A^2=a_1^2+a_2^2-a_3^2-a_4^2$$ After many hours, I found a ...
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### Show all ways 365 can be written as the sum of 2 or more different perfect square numbers? [closed]

I need help with this problem please explain all the appropriate steps.
Well, let's say I have a function $f:\mathbb{R}\to\mathbb{R}$. This function is a polynomial of degree three under a square root sign, which means that is in the form of: $$f(x):=\sqrt{ax^3+bx^2+cx+d}... 3answers 73 views ### Find perfect square ends with 9009 I am trying to solve the following problem. Find perfect square which last 4 digit is 9009 The solution in the textbook starts as follows. Let x be the one we want to obtain. Then x^2 = ... 0answers 75 views ### Proof of the square root method by long division My attempt: Let x  be a number such that$$(y+e)\le (a)^{\frac{1}{2}}$$where a is the number whose square root we are trying to find and e>0 so that y+e is close to square root of a. ... 2answers 536 views ### For any k \gt 1, if n!+k is a square then will n \le k always be true? In Dabrowski's paper, he showed that it would follow from the abc conjecture that the equation$$n!+k=m^2 has a finite number of solutions $n, m$ for any given $k$ which was my motivation to find ...
This is probably a very silly question, but say I have $∥f∥_1^2$ (I'm trying to prove an inequality) and I square root it, do I get $-{∥f∥_1}$ and $∥f∥_1$ or just $∥f∥_1$ due to the definition of the ...