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

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4
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0answers
42 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
5
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3answers
77 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...
1
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2answers
50 views

Square numbers in the form $1+4y$

I want to solve the equation $y+x=x^2$: $$ x^2-x-y=0 \\ x_{1;2}=\frac{1\pm \sqrt{1+4y}}{2} $$ However I want the solutions to be only natural numbers; the question then turns to find values of $y$ ...
1
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0answers
36 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
-4
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2answers
60 views

Find all $n$ such that $n^2+3^n$ is a square number [closed]

Find all $n$ such that $n^2+3^n$ is a square number .
2
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2answers
63 views

Are there any primes for which $a^2 = pb^2 + 1$ does not exist?

The smallest solution to the above equation for various primes are: $(p=2)$ $3^2 = 2*2^2 +1$ $(p=3)$ $2^2 = 2*1^2 +1$ $(p=5)$ $9^2 = 5*4^2 +1$ $(p=7)$ $8^2 = 7*3^2 +1$ Is there at least one ...
12
votes
4answers
232 views

Prove or disprove that $8c+1$ is square number. [on hold]

Let $a,b,c$ be positive integers, with $a-b$ prime, and $$3c^2=c(a+b)+ab.$$ Prove or disprove that $8c+1$ is square number.
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2answers
46 views

If we do not know a number's factors, what is the algorithm (if there is one) to write it as a difference of two squares?

For example, if we have a number like 29873412895, is there an algorithm that can find it as a difference of two squares? Or must you need the factors of the numbers? And what might be the algorithm? ...
-2
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0answers
19 views

Squares, Divisibility, and Fundamental theorem of arithmetic [duplicate]

I want to prove that if $a^2 | b^2$ then $a|b$. Is there an easy way to do this without using the fundamental theorem of arithmetic?
1
vote
2answers
112 views

Is there an algorithm for writing an integer as a difference of squares?

For example, if we have $36$, is there an algorithm to determine that it may equal $10^2-8^2$? What if we blow up the number to something like $492709612098$? Can it be written as the difference of ...
0
votes
2answers
49 views

Please help me solve for $L-L^2$?

Unfortunately, I don't know any basic maths, and I need to solve the following equation for $L$ using the intercept of my graph: $$\begin{aligned} intercept & = \frac{2L}{\pi^2 + L(L - ...
4
votes
0answers
98 views

When is $8x^2-4$ a square number?

I asked an earlier question on when $32x+32$ is a square number (here) and I got a very clear answer. Now I am looking to solve for which $x$ the equation $8x^2-4$ results in a square number. When I ...
6
votes
3answers
147 views

When is $32x+32$ a square number?

I am trying to find out for what values for $x$ does the function $f(x)=32x+32$ return a square number? I found that this is the case for at least: $x \in ...
-1
votes
0answers
49 views

Very tentative proof that the terms in Beal's Conjecture must not be squares?

I'm a high-school student, so please point out my mistakes accordingly. Thanks! Alright. So: $$a^x + b^y=c^z$$ And if x, y, and z are over 2 then $$a^2a^m+b^2b^n=c^2c^o$$ m, n, and o of course ...
0
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2answers
41 views

Proof that irrational coprime square root sums and products are always irrational?

I probably phrased it very bad. This is what I mean: $$\sqrt{x} + \sqrt{y} \neq R$$ x and y being non-square coprime natural numbers. And: $$\sqrt{xy} \neq R$$ x, y, AND R being coprime. Let's try to ...
0
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7answers
52 views

Square rooting to make quadratic function

If I have $x^4 - 34x^2 + 225 = 0$, is it not possible to to square root both sides of the equation so that I now have $x^2 - 34x + 15$? If this is true, then how would I go about solving the equation ...
2
votes
1answer
38 views

Square of One more than the Larger Part

This isn't really much of an algebra problem but different people are giving me different interpretations of " Square of One more than the Larger Part ". I've been given " Separate 17 into two parts. ...
20
votes
0answers
190 views

Can a number be equal to the sum of the squares of its prime divisors?

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ then define $$f(n):=p_1^2+\cdots+p_k^2$$ So, $f(n)$ is the sum of the squares of the prime divisors of $n$. For which natural numbers $n\ge 2$ do we have ...
3
votes
0answers
93 views

Very tentative proof of Beal's Conjecture?

I'm a high school student, so please point out my mistakes nicely and in layman's terms :) Thanks! Ok. Beal's Conjecture: If $$a^x+b^y=c^z$$ where $a$, $b$, $c$, $x$, $y$, $z$ are whole numbers; $x, ...
5
votes
3answers
66 views

Is it possible to have $a^2 + b^2 = c^2 + 1$ for $a$, $b$, $c$ being coprime integers?

As stated above. I'm working on a possible proof. It appears that $$(b+1)(b-1)=(c+a)(c-a)$$ That's where I'm stuck. Any help please? A clear, simple proof desired, thanks!
2
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4answers
116 views

I call them squares. They called them arrays. What do they mean?

So I was in C++, and we had third graders come today to play our programs. Whilst the others just drilled them with problems, my game was subtract a square. It was fun watching them discover that ...
1
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2answers
78 views

Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
6
votes
2answers
86 views

When is the sum of divisors a perfect square?

For $n=3$, $\sigma(n)=4$, a perfect square. Calculating further was not yielding positive results. I was wondering is there a way to find all such an $n$, like some algorithm? We know that if ...
0
votes
2answers
33 views

Continued addition and under rooting of 12

$\sqrt{(12 + \sqrt{12......})}$ and so on.... How do I find its answer? This is a question in our class VII mats book. P.S. - Answer is 4
0
votes
2answers
48 views

Square Root of $320$

Given, $$\sqrt{5} = 2.236$$ $$\sqrt{320} = 2^3 \times \sqrt{5} = 8 \times 2.236 = 17.888$$ This is the explanation provided in my school book. Could someone please elaborate ? Thanks in ...
1
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3answers
100 views

Is $\frac{a^2+b^2}{2}=c^2$ possible?

I am looking for an integer solution to the equation: $$\frac{a^2+b^2}{2}=c^2(a\neq b\neq c)$$ That is a square number that is the mean of two other square numbers, is this possible? And if so please ...
5
votes
1answer
79 views

Prove $(8k)^{8k}+(8k+1)^{8k+1}$ and $(8k+1)^{8k+1}+(8k+2)^{8k+2}$ are never perfect squares

Prove $$(8k)^{8k}+(8k+1)^{8k+1}\ \ \text{ and } \ \ \ (8k+1)^{8k+1}+(8k+2)^{8k+2}$$ are never perfect squares ($k\ge 1$). mod $8$ gives $1$ for both, which is a quadratic residue, so doesn't ...
0
votes
1answer
59 views

Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.

I successfully thought of bounding our expression examining consecutive squares that attain values close to it, and this led to the solution I'll post as an answer, which was the one reported. ...
10
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1answer
75 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+$ such that $a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb ...
-3
votes
2answers
176 views

square of digits - why does it always contain 1 or 89 [closed]

I attempted project euler problem 92, while I passed it, my solution works, but had just...awful performance. So I would like to try again tomorrow. In the meantime understanding why the iteration ...
4
votes
1answer
24 views

Square Roots: Variables with Exponents.

Alright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?
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2answers
27 views

Multistep Equation with Square Root Confusion

Alright, so I have $4 * \sqrt{3} = \sqrt{x}$ So I squared the entire equation to get $$16 * 3 = x$$ $$x = 48$$ Is this correct? Or do I only square the $\sqrt{3}$ part on the left side of the ...
2
votes
2answers
58 views

How many numbers smaller that $N$ can be written as a sum of two square numbers?

We define $$a_N =\# \{ n \leq N, \exists (n_1,n_2) \in \mathbb{N}^2, n = n_1^2 + n_2^2 \}.$$ Can we have the exact value of $a_N$, or at least an asymptotic behavior such as $$ \alpha N \leq a_N \leq ...
1
vote
1answer
82 views

Fermat's Little Theorem and prime divisors

Let $a,b\in\Bbb N$ and $a+b$ be an even number. Assume $a^2 - b^2 - a$ is an exact square, say $c^2$. Let $m = \frac {a+b}2$ and $n = \frac {a-b}2$. Then, $$(4m-1)(4n-1) = 4(4mn-m-n) + 1 = ...
3
votes
3answers
55 views

Integral intersections between quadratic sequences

How can I find the integer solutions to: $$ x^2=\frac{1}{2} n (n+1) $$ By brute force I have found the solutions (6,8) (35,49) and (204,288) but then it gets harder. Note that the perfect squares ...
3
votes
1answer
38 views

Is it true that every sufficiently large positive integer can be written as a sum of a square free number and a perfect square ?

Is it true that $\exists k \in \mathbb Z^+$ such that every integer $n >k$ can be written as a sum of a square free number and a perfect square ?
1
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3answers
145 views

Solve $ \left(\sqrt[3]{4-\sqrt{15}}\right)^x+\left(\sqrt[3]{4+\sqrt{15}}\right)^x=8 $ [closed]

I don't know what can I substitute for $x$ so that equation becomes satisfied. Any assistance will be greatly valued. Thanks!
0
votes
0answers
39 views

Define the concept of square root (mod n)

So i was looking at some of the past papers of my module, the new module has different topics slightly.I was wondering how would you solve this since we didn't cover such topic this year Define the ...
0
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2answers
57 views

The diophantine equation $z^2=a^2+bx^2+cy^2$

Is there a way to obtain (enumerate) the integer solutions $(x,y,z)$ of the following quadratic Diophantine equation $z^2=a^2+bx^2+cy^2$ where $a$ is an integer and $b, c$ are positive integers? I ...
0
votes
0answers
25 views

Squareclasses in transcendental extensions of the p-adics

Let $p$ be any prime and $k = \mathbb{Q}_p$. The structure of the square class group $k^*/k^{*2}$ is well known. It has four or eight elements depending on whether $p$ is odd or not. If we set $K = ...
4
votes
1answer
51 views

$A$ is a sum of two postive integer squares?

if $x,y,z,w$ be postive integer,and such $x^2+y^2$ is prime number,and $A=\dfrac{w^2+z^2}{x^2+y^2}\in N^{+}$ show that $A$ is a sum of two postive integer squares? maybe ...
1
vote
1answer
43 views

Evaluation of infinite square roots

The question is: Evaluate in simplest form:$\sqrt {2013+2012 \sqrt {2013+2012 \sqrt {2013+2012 \sqrt {...} } } }$ Supposing let "x" be $\sqrt {2013+2012 \sqrt {2013+2012 \sqrt {2013+2012 \sqrt ...
0
votes
2answers
65 views

calculate the intersection of two number series

I have a series of numbers. It is in the form of a parabola. This series is guaranteed to have at least one perfect square within it (edited I thought there was only one). The second series is also a ...
3
votes
1answer
42 views

Prove about prime numbers obtained from certain sums of squares of an integer $n$

I would like to ask for a prove about an observation I did regarding the sums of squares and prime numbers (in another question here), or a counterexample of it. My capabilities to do this kind of ...
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votes
2answers
90 views

How to check if $2$ is a square $\mod 3$?

I don't think I can use the Legendre or Jacobi symbol here because $2$ is an even prime. I'm not sure I've learned methods to deal with $2$ even though I know how to use quadratic reciprocity, it only ...
0
votes
1answer
43 views

Show that the sum of the first $n$ natural numbers is a perfect square for infinitely many $n$

Show that the sum of the first $n$ natural numbers is a perfect square for infinitely many $n$ The question doesn't make any sense to me. Any help is appreciated
0
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2answers
64 views

Suppose that $n \in \mathbb{Z}$. Prove that if $n^2 + 1$ is a perfect square, then $n$ is even.

This is a homework problem that I cannot figure out. I have figured out that if $n^2 + 1$ is a perfect square it can be written as such: $n^2 + 1 = k^2$. and if $n$ is even it can be written as ...
7
votes
2answers
96 views

Find $p,q$ s.t. $2q^2-p^2=\Box$ and $2p^2-q^2=\Box$

Problem. Find all integers $p,q$ such that $2q^2-p^2$ and $2p^2-q^2$ are perfect squares. I think this is only true when $p=\pm q$ but I have not been able to prove it. One approach I tried is ...
0
votes
2answers
58 views

Multiplying two expressions containing perfect squares to get another perfect square

Is it possible to multiply a perfect square by the previous square plus one and get another perfect square? An example that doesn't work: $$6^2 (5^2 + 1) = 936 \ne n^2$$
24
votes
14answers
947 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's ...