Questions tagged [square-numbers]

This tag is for questions involving square numbers. A non-negative integer $n$ is called a square number if $n = k^2$ for some integer $k$. Consider using with the [elementary-number-theory] or [number-theory] tags.

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How to show that $16n^2 + 16n+1 \neq m^2$ with $n, m \in \mathbb{N}$?

How to show that $16n^2 + 16n+1 \neq m^2$ with $n, m \in \mathbb{N}$ ? I already know that $m$ needs to be an odd number because: $16n^2 + 16n + 1 \equiv 1 \mod{4}$ I can complement the square by: ...
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4answers
54 views

Solutions to $k$ when $2^k n^2 + 2^k n + 1$ is never a perfect square.

I need to find possible values of $k$ with $k \in \mathbb{N}$ such that for any $n \in \mathbb{N}$ the equation $2^k n^2 + 2^k n + 1$ will never be a perfect square. So, I thought, maybe for even $k$ ...
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1answer
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count number of a solution of an integer equation

How to count all couple of strictly positive integers (u,v) such that the following expression also lends an integer? $$\frac{v}{u}\cdot \left[2v-\sqrt{\left(2v\right)^2-u^2}\right]$$ with \begin{...
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2answers
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Secret pattern of irrational numbers

Are their any pattern in irrational numbers?, I know that there's been no hint as to the appearance of their infinite digits, but I discovered a pattern within themselves $\sqrt1 = 1$ $\sqrt2 = 1.414.....
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1answer
46 views

Could square minus square be a square? [closed]

I wondered if a square minus a square could be a square ? When I put question into equation, I have ...
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1answer
138 views

What is the sum of this modulo series?

Given only a large integer upto $ 10^{18} $ , what can be an efficient way to calculate $$ \sum_{k=1}^{\left \lfloor \sqrt{N} \right \rfloor} \left ( N \ mod \ k^{2} \right ) $$ Note that for a ...
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1answer
45 views

Is this a new method of finding a square of any number?

e.g. number $5$. $5^2=25;$ If I add up the first $4$ even numbers and a half of the $5$th even number, we get $25$. $2+4+6+8+5=25$ So generally speaking, the sum of first $n-1$ even numbers and the ...
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0answers
36 views

Square Modular Summation

I have been working on modular arithmetic, I have tried to find the summation $\sum_{i=1}^N N\pmod {i^2}$ without actual calculation i.e. a shorthand notation or value of this summation for any ...
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1answer
104 views

Find a formula for all pentagonal numbers which are also square numbers. [closed]

I can get the formula for n-th pentagonal number is $P_n=\frac{3n^2-n}{2}$, but I do not know how to get the formula which is also square number. enter image description here
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2answers
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Proving a quadratic equation has no integral roots

Question: Show that the quadratic equation $x^2-7x-14(q^2+1)=0$ where $q$ is an integer ,has no integral real roots. My approach : Let for any integer $x$ the quadratic equation $=0$, Then $x(x-...
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1answer
45 views

Show that if n is an even perfect number then n is not the sum of two squares. [closed]

A perfect number is a positive integer that is equal to the sum of its proper divisors, and all perfect numbers are even. A sum of two squares is an integer that is the sum of two squares integers.
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1answer
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Insights for proving $(x^2-1) = l(l+1)(l+2)(l+3)$ [closed]

I am looking for some insights on how to prove that for every integer $l$, there exists an integer $x$ such that: $$x^2-1 = l(l+1)(l+2)(l+3).$$ So far, what I did was: $$x^2-1=(x-1)(x+1).$$
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4answers
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$100-50\sqrt3$ to be a perfect square

If we want $100-50\sqrt3$ to be a perfect square $(a+b)^2=a^2+2ab+b^2$, most likely it is the case that $-25\sqrt3$ corresponds to $ab$. I can't approach the problem further. Can you give me a hint?
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1answer
117 views

Finding all four-digit perfect squares of the form $XXYY$ [closed]

Can you find a four digit number of the form $XXYY$ using only mathematical tools (without a computer) where the first two digits are same ($XX$) and the last two digits are same ($YY$), and the ...
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3answers
45 views

Find all 6-digit squares which are the concatenation of three 2-digit squares

I am looking to find perfect squares $\overline{abcdef}$ with the property that $\overline{ab}$, $\overline{cd}$ and $\overline{ef}$ are perfect squares. I ran a quick program to find that the only ...
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1answer
57 views

Number Theory Prove a complete square 1,11,111 [closed]

I have to prove that every number in the series 11,111,1111 ... Is not a complete square, Can you give me a clue how to do this?
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$x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square

Determine 4 distinct $A, B, C, D$ for any rational $x$ such that either $x^3+Ax+B$ is not a square number but $x^3+Cx+D$ is a square. or, $x^3+Cx+D$ is not a square number but $x^3+Ax+B$ is a ...
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0answers
56 views

How do you find the closest square number to another number without using a calculator

Say we try to find the closest square number to 26. we already know the closest square number is $25$. However, how do I calculate out 25? Because, if I try to prime factorize it like so: $\...
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1answer
53 views

How to prove this by induction? Having problem in $k+1$ th step.

It is given that $$ -\sqrt{ \frac{8}{3} } \lt x \lt \sqrt { \frac{8}{3} } ~~~~~~~~~~~~\left( x \in \mathbb Q \right)$$ And we want to ...
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1answer
203 views

Prove n is a perfect square if and only if $n^7$ is a perfect square, n $\in$ natural numbers [duplicate]

Proving one way is very simple If n is a perfect square then n = $a^2$ $n^7$ = $a^{14}$ $n^7$ = $a^7$($a^7$) which is obviously a perfect square for some integer a. It's the if $n^7$ is a perfect ...
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2answers
60 views

What is $\sqrt{x^2}$? [duplicate]

At first the question of what $\sqrt{x^2}$ is seems silly. It looks like $x$. And for $x \in \mathbb{R}^+$ it is. However, for $\mathbb{R}$, I'm not sure. I can think of 3 answers: $abs(x)$ $\{x,-x\}...
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0answers
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Proving technique (multiples) [duplicate]

If a and b are integers, if a^2 is a multiple of b^2, will a then become a multiple of b? I feel it's true , 1,4,9,16,25,36,49,64.... I tried many. But I forgot how to formally prove it, how to ...
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1answer
57 views

Infinitely many squares of form 50^m - 50^n? [closed]

I wanted to solve this problem. You have to prove that there are infinitely many square numbers of the form $(50^m - 50^n)$ (and no square numbers of the form $(2020^m + 2020^n)$ with $m$ and $n$ ...
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2answers
77 views

Find all positive integers x such that $x^4-8x+16$ is a full square

Find all positive integers x such that $x^4-8x+16$ is a full square. I presented it as $(x-4)^2+x^4-x^2$ and found that $x = +1$ satisfy the condition. Then I equalized it $(x-4)^2+x^4-x^4=a^2+2ab+b^...
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1answer
64 views

Prove that there are no integer solutions to $ c^2 = 2018^a + 2018^b $

A friend of mine gave me this puzzle and I want to solve it, turns out its harder than I expected. I tried to prove this by contradiction, so let's just assume there is an integer solution. The first ...
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1answer
44 views

Tribonacci Numbers Satisfying Certain Conditions

Here is my question that I tried to solve: Let $(a_n)_{n \in \mathbb{N}}$ and $\displaystyle \frac{1}{1-x-x^2-x^3}=\sum_{n=1}^\infty a_n x^n$ $ \quad $ holds. According to above find all positive ...
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3answers
89 views

What is the term for a shape that has uniform dimensions? [closed]

Is there a term to describe a shape that has uniform dimensions? Squares and circles would fall into this category for 2D shapes, as they are as wide as they are tall. Rather than ellipses or ...
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4answers
87 views

Without a calculator, prove: $7^{1/2} + 7^{1/3} + 7^{1/4} < 7$ [solved]

I need to prove $$7^{1/2} + 7^{1/3} + 7^{1/4} < 7$$ I have reached the stage $$1 + 7^{-2/12} + 7^{-3/12} < 7^{1/2}$$, but don’t know if I’m on the right track or what to do next. Thanks Thanks ...
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2answers
91 views

At least one perfect square in $[S_n,S_{n+1}]$

Given $\{a_n\}$ a sequence of real numbers such that $a_1 > 1$ and $a_{n+1}-a_n\geq 2$, with $S_n=a_1+a_2+...+a_n$, prove that for any $n\geq 3$, there is at least one perfect square in the range $[...
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1answer
51 views

Sum of squares compared to square of sum, divided by number of elements being summed

Let $M, S > 0$ such that $M < S$, where $M$ is a positive integer and $S = \sum_{k = 1}^{M} a_k$, and each $a_k$ is a positive integer. Is it always the case that $\frac{S^2}{M} = \frac{\big(\...
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1answer
129 views

Solving the sequence $a_{n+1}=a_{n}a_{n-1}+\sqrt{(a_n^2-1)(a_{n-1}^2-1)}$: proving that $2+2a_n$ is a perfect square

Question: Let $a_1=a_2=97$ and $a_{n+1}=a_{n}a_{n-1}+\sqrt{(a_n^2-1)(a_{n-1}^2-1)}$ for $n>1$. Prove that (a) $2+2a_n$ is a perfect square, and (b) $2+\sqrt{2+2a_n}$ is a perfect square. I ...
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1answer
27 views

Find the 5 digits number that arrange from x,x+1,x+2,3x,x+3 so that the number is a perfect square

I'm having a hard time dealing with this problem and here is my approach: The perfect square of 5 digits number must be a 3 digits number ( I put it as a,b,c ) $abc^2$ get us a 5 digits number that ...
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1answer
339 views

Conjecture : an odd perfect square $n>1$ raised to the $m$-th power is never divisible by the sum of $n$'s divisors

This is a conjucture that I created : Let $\,n = (2k+1)^2 \,\, $with $k\in \mathbb{N}$ and so $n>1$, and let $$\,\,A = \sum_{d \in \mathbb{N}; \ d|n} d.$$ Then $n^m$ is never divisible by $A$ ...
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1answer
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$x_{n + 3} = \frac{(n + 1)(n^2 + n + 1)}{n}x_{n + 2} + (n^2 + n + 1)x_{n + 1} - \frac{n + 1}{n}x_n, n \ge 1$, $\sqrt{x_n} \in \mathbb N, n \ge 0$.

Given sequence $(x_n)$ such that $x_0 = 0, x_1 = 1, x_2 = 1, x_3 = 4$ and $$x_{n + 3} = \frac{(n + 1)(n^2 + n + 1)}{n}x_{n + 2} + (n^2 + n + 1)x_{n + 1} - \frac{n + 1}{n}x_n, \forall n \ge 1$$ Prove ...
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3answers
194 views

Sum of two perfect squares is also a perfect square. Proof that one of these numbers is divisible by 3

At first, I tried to proof by contradiction: I considered two numbers that are not divisible by $3$. Than I tried to write consecutive perfect squares that are divisible by $3$ to see some pastern, ...
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2answers
151 views

Find all $n\in\mathbb N$ such that $10^n-6^n$ is a perfect square

EDIT: I have edited the title. Thanks to Dietrich Burde, we now have that there is no $n\geq4$ such that $a^n-b^n$ is a perfect square for coprime $a,b$. It shows for $5^n-3^n,7^n-3^n$ and $10^{2m}-6^{...
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1answer
69 views

Square numbers and divisibility

I'm trying to prove the following statement: If $kn+1$ and $(k+1)n+1$ are both perfect squares, then $2k+1$ divides n, where k and n are positive integers. Case $k=1$: Put $n+1=x^2$, $2n+1=y^2$. ...
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3answers
161 views

Does there exist $n\in\mathbb{N}$ such that $5^n-2^n$ is a perfect square?

I checked up to $n\leq 100000$ then found no example. So I suspect that there doesn't exist $n\in\mathbb{N}$ such that $5^n-2^n$ is a perfect square. Then I tried to prove by modular arithmetic, but ...
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1answer
30 views

gnomes and lights vs. sieve of Eratosthenes

I was riddled a riddle the other day - a series of gnomes flips a series of light switches. The first gnome flips every switch, the second gnome flips every second switch, the third gnome flips every ...
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3answers
153 views

Prove that $A^2=a_1^2+a_2^2-a_3^2-a_4^2$ for all integers $A$.

Prove that: For every $A\in\mathbb Z$, there exist infinitely many $\{a_1,a_2,a_3,a_4\}\subset\mathbb Z$ given $a_m\neq a_n $ such that $$A^2=a_1^2+a_2^2-a_3^2-a_4^2$$ After many hours, I found a ...
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1answer
41 views

Digits & Squares

If $\overline{abcd} = (\overline{ab} + \overline{cd})^2$ and only $c$ can be $0$, find the sum of all possible values of $\overline{abcd}$. $(A) 13850$ $(B) 14051$ $(C) 14742$ $(D) 14851$ $(E) ...
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2answers
198 views

A generalization of the (in)famous IMO 1988 problem 6: If $\frac{a^2 + b^2 - abc}{ab + 1}$ is a positive integer then it is a square.

This question is motivated by the famous IMO $1988$ problem $6$. Is the following true? Let $a,b$ be positive integers and $c \ge 0$ be a non-negative integer. If $\dfrac{a^2 + b^2 - abc}{ab + 1}$ ...
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1answer
236 views

Proof that $a^2 + b^2 + ab$ and $a^2 + b^2 - ab$ can not both be perfect squares

If $a$, $b$ are co-prime integers, how can I prove that $a^2 + b^2 + ab$ and $a^2 + b^2 - ab$ cannot be perfect squares? I know that perfect squares should be capable of expression in the form $a^2 + ...
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3answers
114 views

Show all ways 365 can be written as the sum of 2 or more different perfect square numbers? [closed]

I need help with this problem please explain all the appropriate steps.
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1answer
97 views

Mathematical shortcut to determine if a number is a integer

Well, let's say I have a function $f:\mathbb{R}\to\mathbb{R}$. This function is a polynomial of degree three under a square root sign, which means that is in the form of: $$f(x):=\sqrt{ax^3+bx^2+cx+d}...
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3answers
73 views

Find perfect square ends with 9009

I am trying to solve the following problem. Find perfect square which last 4 digit is 9009 The solution in the textbook starts as follows. Let $x$ be the one we want to obtain. Then $x^2 = ...
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0answers
75 views

Proof of the square root method by long division

My attempt: Let $x $ be a number such that $$(y+e)\le (a)^{\frac{1}{2}}$$ where $a$ is the number whose square root we are trying to find and $e>0$ so that $y+e$ is close to square root of $a$. ...
14
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2answers
536 views

For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true?

In Dabrowski's paper, he showed that it would follow from the abc conjecture that the equation $$n!+k=m^2$$ has a finite number of solutions $n, m$ for any given $k$ which was my motivation to find ...
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2answers
35 views

Square rooting results

This is probably a very silly question, but say I have $∥f∥_1^2$ (I'm trying to prove an inequality) and I square root it, do I get $-{∥f∥_1}$ and $∥f∥_1$ or just $∥f∥_1$ due to the definition of the ...

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