# Tagged Questions

A non-negative integer $n$ is a square number if $n = k^2$ for some integer $k$.

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### Infinitely many squares of the form $3a^2 + 1$

I am working on the following proof. Prove that $3a^2 + 1$ (as $a$ ranges over the integers) produces infinitely many squares. Proof: We first note that all square numbers can be represented as ...
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### How can I solve this recurrence relation for generating triangle-squares?

$$N_k = 17N_{k-1} + 6(8N^2_{k-1} + N_{k-1})^{1/2} + 1$$ $$k\geqslant 1$$ I'm trying to convert a recurrence relationship for producing triangle square-numbers into a closed-form expression in terms of ...
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### characterisation of $n$ as prime using min values of $x$ such that $nx+1$ or $nx$ is square

Let $n\ge 5$ be an odd integer and $k\ =\ \min\{x\in\mathbb{N}\colon nx+1\text{ is a perfect square}\}$ $l\ =\ \min\{x\in\mathbb{N}\colon nx\text{ is a perfect square}\}$ Prove that $n$ is a prime ...
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### Proving that $2^{2a+1}+2^a+1$ is not a perfect square given $a\ge5$

I am attempting to solve the following problem: Prove that $2^{2a+1}+2^a+1$ is not a perfect square for every integer $a\ge5$. I found that the expression is a perfect square for $a=0$ and $4$. ...
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### Prove that $4m + 1$ is a perfect square if $\{ \sqrt {n + \sqrt n}\} = \{\sqrt m\}$

Let $n,m \in \mathbb{N}-\{0\}$ so that $\{ \sqrt {n + \sqrt n}\} = \{\sqrt m\} \tag1$ Prove that $4m + 1$ is a perfect square. ($\{x\}$ is the fractional part of $x$) No idea how to start. ...
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### How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?

I know a little about Euler's totient function that counts integer less than $N$ that are relatively prime to $N$. But I don't know how to modify the function for perfect square numbers, or maybe ...
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### Is difference of two consecutive sums of consecutive integers (of the same length) always square?

I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
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### Finding a pattern regarding perfect cubes.

For the purpose of this question and the patterns pointed out, $0$ is not a perfect square (some think it is, others no). The first perfect squares are $1$, $4$, $9$, $16$, $25$ and so on. The ...
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### Deciding if a number is a square in $\Bbb Z/n\Bbb Z$

I am looking for a systematic way of deciding if a given number is a square in $\Bbb Z/n\Bbb Z$. E.g. is $89$ a square in $\Bbb Z/n\Bbb Z$ for $n\in \{25,33,49\}$? Brute-forcing it would take too ...
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### Prove That $n(n+1)$ Can Never Be a Square Number by Showing the Atleast One of Exponents in the Prime Power Decomposition Isn't Even

Can someone show me how to prove that when $n>0$, $n(n+1)$ can never be a square number by demonstrating at least one of the exponents in the prime power decomposition is not even? Here's what I ...
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### Why are the solutions of the equation different? : $x=2 => x^2=4 => x=±2$

If I define the variable $x$ as $x=2$, then $x^2=4$. But the solutions of $x^2=4$ are $±2$(two solutions). I defined what the variable $x$ is, then why are the solutions for the equation $x^2=4$ two, ...
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### For which integers $q \ge p\ge 1$ with $q^2-2p^2=2$ is $2p^2+1 \pm pq$ an integer square?

The title says it all… I’m looking to prove (in an elementary way, if possible) the following question: Conjecture: If $q$ and $p$ are positive integers such that $q^2-2p^2=2$ and $2p^2+1 \pm pq$ is ...
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### Existence of a Repeating Divisor

I have $n$ integers $a_1, a_2, a_3, .., a_n$ let $X = a_1*a_2*a_3*...*a_n$. I want to know a single integer $F$ such that $F^2$ divides $X$. It is told that there will be atleast one such $X$ and ...
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### No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
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### What is the first square in the sequence $4729494n+1$?

Today I found a strange phenomenon that I want to ask about. If $$f(n)=4729494n+1,$$ is square, where $n$ is positive integers. Then I found $n=4729492$, because $$f(4729492)=4729493^2$$ In fact,...
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### $n=a^2-b^2$ iff $n \not\equiv 2(\mathrm{mod\ }4)$

I have to show that $n=a^2-b^2$ iff $n\not\equiv 2$ (mod $4$). Where $a$, $b$ are integers. I already got the explicit $(a,b)$ if $n\not\equiv 2$ (mod $4$). However, I am stuck with the other ...
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### Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square

$\textbf{Question.}$ Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square One thing which I observed is the following. Since $5 \equiv 1 \pmod{4}$, this ...
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### Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$.

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$. I tried taking all possible combinations $\bmod 3$ and $\bmod 4$ and it has a solution only ...
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### Find the $m$ such $a_{n+1}=a^5_{n}+487$ [closed]

Let $\{a_{n}\}$ be a sequence of positive integers, and suppose $a_{0}=m$. Further, $\{a_{n}\}$ satisfies $$a_{n+1}=a^5_{n}+487.$$ Find $m$ so that this sequence consists of square numbers for as long ...
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### How do I prove that $\sqrt{9+4k^2}$ holds integer value only for $k=0$ and $k=2$?

I've faced that sort of a problem while solving some other problem and it made me stuck for a while. It's vital to me to prove that for any other integer $k$ there can't be an integer output, i.e. a ...
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### Consecutive Square Numbers [closed]

The difference between the squares of two consecutive numbers is $23$. What are the two numbers?
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### Pattern involving squares, primes, and remainders

I ran across a really neat pattern, wholly by accident. In advance, my questions are: Has this been discovered before? If so, where can I learn more about it? Why does this pattern work? Now for ...
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### Perimeter/footprint and square meters

I'mIf a room is 8x8 meter the perimeter will be 32m, and 64m2. If add 2m on two sides and subtract 2m on two sides the room will be 10x6m. The perimeter, and footprint, will be the same for both rooms ...
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### Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function.

The following is an exercise in my textbook on complex analysis: Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function. ...
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### Prove that $2^n+3^n$ is never a perfect square

My attempt : If $n$ is odd, then the square must be 2 (mod 3), which is not possible. Hence $n =2m$ $2^{2m}+3^{2m}=(2^m+a)^2$ $a^2+2^{m+1}a=3^{2m}$ $a (a+2^{m+1})=3^{2m}$ By fundamental ...
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### Square Number Problem

If one is given a set of digits, such as 3352, is there a simple way of finding the square numbers closest to 0 that would begin with the given digits?
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### Do you write plus/minus if a variable squares equals the square root of a number?

For example, if I have $x^2 = \sqrt{49}$. I know that $7$ is the number, but as my final answer, do I write that $x = +\sqrt{7}$ and $-\sqrt{7}$ or just $x = \sqrt{7}$?
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### Sum of two consecutive squares equal square

$N^2 + (N+1)^2 = K^2$, find all solutions for $N<200$ I have done some factoring and also realized that $K=[n\sqrt{2}]+1$ in eventual solutions, where $[x]$ denotes the greatest integer less than....
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### Is it correct to say that Square Root of 0.9 is 0.9 itself

When I calculate Square Root of 0.9, it comes around 0.9486832980505138. Though I have heard people occasionally saying that the square root of 0.9 is 0.9 itself. Would it be correct to make a ...
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### Pattern in digits of sums of consecutive squares

I am interested in patterns in square numbers as well as the reasons behind them and I can't seem to figure out (also how to prove) why do the sums of two consecutive squares only end in digits 1, 3 ...
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### hailstone sequence of perfect squares (collatz conjecture)

The Collatz conjecture states: Take any positive integer $n$. If $n$ is even, divide it by $2$ to get $n/2$. If $n$ is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process ...
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### $m^2+n^2$ and $m^2-n^2$ cannot both be squares [duplicate]

I need to show that there aren't any $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both squares. First of all, assume without loss of generality that $m$ and $n$ are co-prime, since otherwise we ...
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### Does a square root come out plus/minus even if there is a negative sign outside?

For example: $-\sqrt{100x^{20}y^{10}}$. Would that give $\pm10x^{10}y^5$ or just $-10x^{10}y^5$?
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### Square Root of $5$ mod $10^{9}+7$ [closed]

$My$ $Current$ $Knowledge:$ We can find it if 5 is a $Quadratic$ $residue$ modulo p and where p is prime and we can check it using $Euler$ $criterion$. I cannot able to find the root(5)mod 1000000007. ...
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### Finding large integer squares - how to decide modulus to calculate.

Say I want to decide if an integer is a square. My integer is rather big so I can't keep it in memory altogether, but I know that it is a product $$f = \prod_{i=1}^nf_i, f_i\in \mathbb{N}$$ Then I ...
### How to prove that $gxyz$ and $g(y-x)$ are perfect squares? [duplicate]
Let $x,y,z$ be positive integers such that $1/x-1/y=1/z$. Let $g=\gcd(x,y,z)$. Prove that $gxyz$ and $g(y-x)$ are perfect squares.