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

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5
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3answers
47 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
votes
4answers
102 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 ...
0
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2answers
54 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
78 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
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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
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2answers
47 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
vote
3answers
96 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
75 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
50 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. ...
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votes
2answers
162 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 ...
-2
votes
0answers
48 views

What is $\sqrt{-1} \times \sqrt{-1}$? [duplicate]

I have a very simple doubt. $\sqrt{-1} \times \sqrt{-1} = -1$ as per the logic $j(1) \times j(1) = -1$. Why not this way ? $\sqrt{-1} \times \sqrt{-1} = \sqrt{-1 \times(-1)} = \sqrt{+1} = 1$ ????? ...
4
votes
1answer
23 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?
1
vote
2answers
26 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
79 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
52 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
35 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
134 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
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0answers
34 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
votes
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
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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
48 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 ...
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vote
1answer
40 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
61 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
88 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
38 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
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2answers
60 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
93 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
933 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 ...
3
votes
2answers
64 views

How to find all positive integers $m,n$ such that $3^m+4^n$ is a perfect square?

How to find all positive integers $m$, $n$ such that $3^m+4^n$ is a perfect square? I have found $m=n=2$ is a solution, but cannot find any other and cannot prove whether there is any other solution ...
2
votes
2answers
78 views

Proving an expression is perfect square

I have this expression I got in one larger exercise: $$\frac{(2+\sqrt3)^{2n+1}+(2-\sqrt3)^{2n+1}-4}{6}\frac{(2+\sqrt3)^{2(n+1)+1}+(2-\sqrt3)^{2(n+1)+1}-4}{6}+1$$ and i need to prove it is perfect ...
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2answers
50 views

What are the properties of the positive real numbers pair $(a,b)$ for which $a b \geq a + b$?

Working set: $\left\{ a,b\in(0,\infty)\right\} $ For example I'm considering these as including/excluding sets: Excluding 1. For any pair of numbers that, $a\in (0,1)$ and $b\in (0,\infty)$, the ...
4
votes
1answer
101 views

The smallest $n$ for which $19n+1$ and $95n+1$ are perfect squares

Find the smallest possible integer $n$ for which $19n+1$ and $95n+1$ are both perfect squares. I somehow managed to show that $n$ is odd but couldn't find any solution for which both of them are ...
2
votes
3answers
117 views

How to determine what numbers are perfect squares without calculator?

Please explain how you can do this?
6
votes
3answers
86 views

Show that $m+3$ and $m^2 + 3m +3$ cannot both be perfect cubes.

Show that $m+3$ and $m^2 + 3m +3$ cannot both be perfect cubes. I've done so much algebra on this, but no luck. Tried multiplying, factoring, etc.
0
votes
0answers
36 views

Can you find squares in this class?

For a problem I am working over, I would like to prove that numbers of the type are not squares $p(l^4+6l^2m^2-3m^4)$ where $p,l,m$ are integers an $p$ prime. I have already found various necessary ...
5
votes
2answers
149 views

A prime number generator algorithm based on $x^2+(x-1)^2$ that generates only primes

I think I could have found a prime number generator algorithm, but still I am not very sure, maybe this is an already known property of perfect square numbers, maybe not, but it looks amazing and I ...
0
votes
1answer
68 views

Is there a primality test based on the sum of squares of the first $n$ natural numbers $\sum_{x = 1}^{n} x^2$?

The Fibonacci and Catalan primality tests are based on the calculation of the congruences of those numbers versus the possible prime $n$ (the rules are different depending on the primality test), and ...
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2answers
44 views

When is this sequence of positive integers a square?

I have two sequences below, and I would like to know for which $n$ the number $k_n$ is a square. $$ \begin{align} k_1 &= 9\\ t_1 &= 1\\ k_{n+1} &= 9k_n + 80t_n\\ t_{n+1} &= k_n + ...
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vote
2answers
43 views

Is $8r+1$ always a square for integer $r$?

Assume that $r$ is an integer. Since either $t$ or $t+1$ is even, $t$ is an integer for any integer $r$. $$ \begin{align} 2r &= t(t+1)\\ 8r &= 4t(t+1)\\ 8r &= 4t^2 + 4t\\ 8r + 1 &= ...
4
votes
1answer
74 views

When is $20q^4-40q^3+30q^2-10q$ a square for positive integer $q$?

For what $q$ is the following polynomial a square? $$ \begin{align} &20q^4-40q^3+30q^2-10q\\ =\:&10q(q - 1)(2q^2 - 2q + 1) &q\in\mathbb N \end{align} $$ I know of two single cases, ...
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vote
4answers
83 views

Prove that there is no perfect square that is congruent to 2 mod 10 and 3 mod 10.

Prove that there is no perfect square that is congruent to 2 mod 10 and 3 mod 10. Can someone tell me how to solve this question. I really can't figure out.
3
votes
1answer
87 views

Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.

Inspired by this interesting question and in order to solve an old problem, I have the following question: Can we construct a strictly increasing sequence $(N_i)_{i\in \mathbb{N}}$, such that for ...
0
votes
1answer
29 views

Application: Sum of Digits

if a five digit number N is such that sum of its digit is 29, can N be square of an integer? Suppose N be abcde, where a+b+c+d+e = 29. Can square of any number less than abcde is equal to abcde ...
9
votes
2answers
499 views

when is $n!+10$ a perfect square?

When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
7
votes
0answers
96 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
1
vote
0answers
61 views

$t$ is a square $\pmod{2^n}$ if and only if $t\equiv 1 \pmod 8$

Show that $t$ is a square $\pmod {2^n} \iff t\equiv 1\pmod{8}$, given that $t$ is odd and $n \ge 3$. I've tried proving forwards using Hensel's lemma, but got stuck.
6
votes
4answers
119 views

Why can an integer written $2$ times in a row never be a perfect square?

It seems to be true for all natural numbers below $1,000,000$. I am really stuck any kind of help will be appreciated!