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

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0
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2answers
62 views

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 ...
0
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0answers
33 views

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 ...
3
votes
2answers
112 views

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 ...
5
votes
1answer
69 views

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 $4$. ...
2
votes
2answers
89 views

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. ...
2
votes
1answer
29 views

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 ...
35
votes
9answers
2k views

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....
0
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2answers
27 views

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 ...
2
votes
3answers
64 views

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 ...
3
votes
2answers
66 views

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 ...
0
votes
3answers
46 views

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, ...
1
vote
2answers
63 views

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 ...
0
votes
1answer
13 views

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 ...
1
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0answers
51 views

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 ...
3
votes
1answer
74 views

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,...
4
votes
2answers
82 views

$n=a^2-b^2$ 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 ...
4
votes
1answer
92 views

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 ...
8
votes
2answers
178 views

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 ...
0
votes
1answer
81 views

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 ...
1
vote
3answers
67 views

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 ...
1
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2answers
51 views

Consecutive Square Numbers [closed]

The difference between the squares of two consecutive numbers is $23$. What are the two numbers?
2
votes
0answers
37 views

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 ...
0
votes
1answer
27 views

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 ...
1
vote
0answers
26 views

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. ...
10
votes
4answers
162 views

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 ...
0
votes
2answers
33 views

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?
2
votes
3answers
55 views

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 $x = \sqrt{7}$?
1
vote
1answer
85 views

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....
1
vote
1answer
65 views

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 ...
0
votes
2answers
37 views

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 ...
1
vote
1answer
61 views

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 ...
3
votes
0answers
70 views

$m^2+n^2$ 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 ...
-1
votes
2answers
49 views

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 $-10x^{10}y^5$?
8
votes
0answers
68 views

Sum of three consecutive cubes equals a perfect square

I have found this problem in an old German textbook: Find all sets of three consecutive integers such that the sum of their cubes is a perfect square. We can write $$S = (x-1)^3 + x^3 + (x+1)^3 = (x-...
-3
votes
3answers
84 views

Calculating square roots without a calculator [closed]

Can anyone help me calculate $\large{\sqrt{\frac{4}{11}}=\sqrt{0.\overline{36}}}$ using the digit by digit method?
4
votes
1answer
67 views

Prove that for $k>1$ $a_n$ is a perfect square

I'm having problems with this exercise. Let $k > 1$ be an integer. We define $(a_n)_{n \in \Bbb N_0}$ as: $$a_0 = 1$$ $$a_1 = 1$$ $$a_{n+2} = (k^2-2)a_{n+1}-a_n-2(k-2)$$ Prove that $\forall n \in \...
3
votes
4answers
67 views

$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers

$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ My attempt $\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$ $(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$ $x+938^2 + 2*\sqrt{x+938^2}*\sqrt{...
2
votes
3answers
74 views

Finding a number $n$ and $k$ such that $nx+k$ will be a perfect square for any two given $x$.

Given two positive integers $x_1,x_2$, is it always possible to find positive integers $n$ and $k$ such that the expression $nx_i+k$ becomes a perfect square for each $i$ ?
2
votes
0answers
24 views

What triples of square-free integers $(r,s,t)$ admit integer solutions $(x,y,z)$ where $rx^2,sy^2,tz^2$ are consecutive integers?

In this post on the consecutive integers $b^2,2a^2,3c^2$, I asked whether the trivial solution $a=b=c=1$ was the only one. At this time, that question appears to have been answered in the affirmative (...
2
votes
2answers
38 views

Counting Squares In a Range

I've been working on a series of programming challenges to work on my math skills, and I came across a solution that I don't know how to explain. The problem: "For ...
2
votes
1answer
51 views

variant of Lagrange's four square theorem using a restricted set of squares

The well-known four square theorem states that any positive integer is the sum of at most four squares. Suppose that, instead of allowing all squares, I only consider the following set of squares: $$ ...
-1
votes
1answer
61 views

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. ...
0
votes
0answers
20 views

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 ...
1
vote
0answers
22 views

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.
1
vote
1answer
96 views

How many perfect squares exist? [closed]

Consider a set of $1985$ positive integers not necessarily distinct. Every number in set can be written in the form $p_1^{{\alpha _1}}p_2^{{\alpha _2}} \cdots p_9^{{\alpha _9}}$ where $p_1,p_2,\ldots,...
0
votes
1answer
81 views

Primes Between Squares of Primes

Is this problem still open? I know that Henri Brocard conjectured that there are at least four primes in the interval between each pair of consecutive squares of primes from nine onward. http://...
1
vote
1answer
64 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n
1
vote
1answer
62 views

Why this book says that $ 2^{1/2} = ±\sqrt{2} $?

Shouldn't it be: $ 2^{1/2} = \sqrt{2} $ ? I know the problem is that there they are working with complex numbers, but I still don't understand. The book is in the link, page 113, when they move ...
1
vote
0answers
32 views

Floor Function and Euler-Mascheroni Constant

I need to know how to express the following function, $\lfloor{\sqrt{c^2-707}}\rfloor^2=y$ $(s.t.$ $c $ and $y$ are in $N)$, analytically. I think http://mathworld.wolfram.com/Euler-...
0
votes
1answer
132 views

Different proofs for $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ [closed]

Different proofs that show $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ cannot be the square of an integer, where n is a natural number.