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

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3
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1answer
21 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
112 views

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

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
23 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
50 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
22 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 ...
1
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1answer
37 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
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2answers
56 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
39 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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2answers
81 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
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1answer
34 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
59 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
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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: $$6^2 (5^2 + 1) = 936 \ne n^2$$
21
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13answers
821 views

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

$\require{cancel}$ 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 ...
3
votes
2answers
58 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
71 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
48 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
98 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
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3answers
95 views

How to determine what numbers are perfect squares without calculator?

Please explain how you can do this?
6
votes
3answers
83 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.
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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 ...
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votes
2answers
137 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
61 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
40 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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2answers
42 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
71 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
82 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
28 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
495 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
91 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 ...
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0answers
60 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
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4answers
112 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!
1
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1answer
29 views

Difference of the difference of consecutive squares

I was playing around with the square numbers and I noticed something: $$\left. \begin{array}{r} &\left. \begin{array}{r} &1^2 = 1\\ &2^2=4\\ \end{array} \right\} \ 3 \\ &\left. ...
0
votes
1answer
27 views

Impossibility of Equation

Prove that there are no solutions to $ k^2 = x^4 + 2x^3 + 2x^2 + 2x + 1 $ in $ \mathbb Z^+$. I have tried a bounding argument so far, placing $k^2$ in between $x^4$ and $(x+1)^4$, but I am unable to ...
2
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0answers
43 views

Numbers in the form $10^n + 1$ with square divisors

Basically, describe every number in the form $10^n + 1$ with square divisors meaning at least one of it's divisors is a square. Of course, there's infinite, but give a general algorithm for finding ...
3
votes
0answers
40 views

A bound on the squares of primes

If $p_n$ is the $n$th prime, what is the best known upper bound on $m$ such that $p_n \cdot p_{m+n} < p_{n+1}^2$?
1
vote
1answer
38 views

$F$ is a field and $f(a) =a^2$ is a permutation of $F$. What is the characteristic of $F$?

Let $F$ be a field satisfying the condition that the function $a \mapsto a^2$ is a permutation of $F$. What is the characteristic of $F$? Any suggestions? This is an excercise from the book: The ...
8
votes
2answers
288 views

If $ab-1,bc-1,ca-1,ab-a-b+c,bc-b-c+a,ca-c-a+b$ are perfect squares, then are $ab+a+b-c,bc+b+c-a,ca+c+a-b$ also perfect squares?

About a month ago, a friend of mine taught me that there exist many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ ...
0
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1answer
48 views

$d(n)$ is odd iff $n = k^2$ [duplicate]

The function $d(n)$ gives the number of positive divisors of $n$, including $n$ itself. For example, $d(25) = 3$ because $25$ has three divisors: $1$, $5$, and $25$. Prove that $d(n)$ is odd if and ...
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3answers
79 views

Why $27^{1/3} = 3\sqrt{27} = 3$? as $27^{1/3} = 9$ and $3\sqrt{27}=3$?

Why $27^{1/3} = 3\sqrt{27} = 3$? as $27^{1/3} = 9$ and $3\sqrt{27}=3$? Link: http://www.mathsisfun.com/algebra/exponent-fractional.html Please Explain
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votes
2answers
96 views

If $x^{1/2}$ is the same as $ \sqrt[2] x$ then why $x^{1/3}$ is not equal to $\sqrt[3] x$?

If $x^{1/2}$ is the same as $\sqrt[2] x$ than why $x^{1/3}$ is not equal to $\sqrt[3] x$ and $x^{1/4}$ not equal to $\sqrt[4] x$ and so on...? Thank you.
2
votes
1answer
74 views

Can someone help me prove that $\tau(n)$ is odd iff $n$ is a perfect square. [duplicate]

Can someone help me prove that $\tau(n)$ is odd if and only if $n$ is a perfect square. So basically I have to prove that $\tau(n)$ is odd iff $n = k^2$ for some integer $k$. $\tau(n)$ is the ...
12
votes
1answer
141 views

Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?

I've been looking for perfect squares that can be represented as $\sum\limits_{k=0}^{n}p^k$. Of course, both $n$ and $p$ should be natural numbers larger than $1$. Searching up to $n=100$ and ...
0
votes
1answer
82 views

polynomial equation solve in full numbers

$x^2+x+41=y^2$ <--solve that in full numbers. I get to that point $(y-sq(x))(y+sq(x)))=x+41$ which imo must be false because this implies that $sq(x)^2$ is equal to $-x$ did I make a mistake ...
0
votes
1answer
47 views

Square Roots with Exponents

I learned about Square roots and with exponents, but not this: The radius $r$ in millimeters of a platinum wire $L$ centimeters long with resistance $0.1$ ohm is $r = 0.059L^\frac 12$. How long is a ...
6
votes
1answer
245 views

Sum of a Sequence of Prime Powers $p^{2n}+p^{2n-1}+\cdots+p+1$ is a Perfect Square

Find all primes p such that $p^{2n}+p^{2n-1}+p^{2n-2}+\cdots+p^{2}+p+1$ is a square for some value of n.
0
votes
0answers
25 views

About primitive roots and non-square-free numbers.

IF the order of $m^2n\pmod p \equiv p-1$ and then there exists $a , b$ elements of integers | $(m^2 n)^a\equiv m^2\pmod p$ and $(m^2 n)^b \equiv n \pmod p$ so $(m^2 n)^{a+b-1} \equiv 1 \pmod p$. ...
0
votes
1answer
31 views

General term of a series that subtracts the square root of every square.

I'm trying to figure out the general term for a series where you subtract from every perfect square number, its square root. So ...