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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109
votes
1answer
4k views

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ ...
93
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7answers
46k views

What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

Problem 6 of the 1988 International Mathematical Olympiad notoriously asked: Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect ...
86
votes
4answers
51k views

Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer

Let $a,b$ be positive integers. When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square. Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two ...
85
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1answer
3k views

Numbers $n$ such that the digit sum of $n^2$ is a square

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the following: $$S\left(9\color{red}{^2}\right)=S(81)=8+1=3\color{red}{^2}$$ $$S\...
60
votes
7answers
2k views

Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.

Let $k$ be a natural number. Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers. I tried to prove this by supposing one of them is a square number and by substituting the corresponding $k$ ...
51
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3answers
6k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
43
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7answers
4k views

Can any number of squares sum to a square?

Suppose $$a^2 = \sum_{i=1}^k b_i^2$$ where $a, b_i \in \mathbb{Z}$, $a>0, b_i > 0$ (and $b_i$ are not necessarily distinct). Can any positive integer be the value of $k$? The reason I am ...
41
votes
6answers
981 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 $f(n)=...
36
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9answers
3k 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....
35
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7answers
6k views

Why is there a pattern to the last digits of square numbers?

I was programming and I realized that the last digit of all the integer numbers squared end in $ 0, 1, 4, 5, 6,$ or $ 9 $. And in addition, the numbers that end in $ 1, 4, 9, 6 $ are repeated twice ...
34
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2answers
1k views

Is $100$ the only square number of the form $a^b+b^a$?

Conjecture $100$ is the only square number of the form $a^b+b^a$ for integers $b>a>1$. In other words, $(a,b)=(2,6)$ is the only solution. Can we prove/disprove this? Observations The ...
33
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1answer
7k views

Math olympiad 1988 problem 6: canonical solution (2) without Vieta jumping

There is a recent question about this famous problem from 1988 on this forum, but I'm unable to respond to this because the subject is closed for me (insufficient reputation). Therefore this new post ...
29
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5answers
11k views

Can $n!$ be a perfect square when $n$ is an integer greater than $1$?

Can $n!$ be a perfect square when $n$ is an integer greater than $1$? Clearly, when $n$ is prime, $n!$ is not a perfect square because the exponent of $n$ in $n!$ is $1$. The same goes when $n-1$ is ...
28
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10answers
14k views

Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer

Please help me with solving this : prove that none of $\{11, 111, 1111 \ldots \}$ is the square of any $x\in\mathbb{Z}$ (that is, there is no $x\in\mathbb{Z}$ such that $x^2\in\{11, 111, 1111, \ldots\}...
26
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1answer
392 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
26
votes
2answers
530 views

What is this pattern found in the first occurrence of each $k \in \{0,1,2,3,4,5,6,7,8,9\}$ in the values of $f(n)=\sqrt{n}-\lfloor \sqrt{n} \rfloor$?

Learning how to generate the Mandelbrot set, I came across the definition of the "escape condition" which is the one that decides the color that is applied to each point of the plane where ...
23
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8answers
5k views

Why are the last two digits of a perfect square never both odd?

Earlier today, I took a test with a question related to the last two digits of perfect squares. I wrote out all of these digits pairs up to $20^2$. I noticed an interesting property, and when I got ...
23
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15answers
1k 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 $$\...
21
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3answers
2k views

Prove that the number 14641 is the fourth power of an integer in any base greater than 6?

Prove that the number $14641$ is the fourth power of an integer in any base greater than $6$? I understand how to work it out, because I think you do $$14641\ (\text{base }a > 6) = a^4+4a^3+6a^2+...
21
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1answer
2k views

Find All $x$ values where $f(x)$ is Perfect Square

Is there a formula, method or anyway to find all positive $x$ integer values (if exists) such that $f(x)$ is Perfect square where $f(x)$ is a quadratic equation? For example if I have the following ...
20
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2answers
217 views

Sequences where $\sum\limits_{n=k}^{\infty}{a_n}=\sum\limits_{n=k}^{\infty}{a_n^2}$

I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form. I then wondered ...
19
votes
1answer
579 views

All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 \...
19
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1answer
3k views

For which $n$ can $\{1,2,…,n\}$ be rearranged so that the sum of each two adjacent terms is a perfect square? [duplicate]

For which numbers $n$ can the sequence $1$ to $n$ be rearranged such that each pair of consecutive terms adds up to a perfect square? Can this be done on the set of natural numbers as well? Integers? ...
18
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7answers
2k views

Subtracting Quarters of Squares Equals Multiply?!

Can anyone explain to me how/why this works (hopefully in mostly layman's terms)? It seems pretty magical to me at the moment. $${{(a+b)^2\over4} - {(a-b)^2\over4}} = a b.$$
17
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2answers
1k views

When is $2^n -7$ a perfect square?

This came up while solving another ENT problem. I want to ask when is: $$2^n -7 \text{ where } n\geq 3$$ a perfect square? Specifically, I also wanted to know what would be the solutions when $n$ is ...
17
votes
1answer
390 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 $p=200$...
17
votes
1answer
474 views

$\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square

Let $x,y\ge 1$ be non-integer real numbers such that $\lfloor x^n\rfloor\lfloor y^n\rfloor$ is a perfect square for all natural numbers $n$. Does it follow that $x=y$? From this question we know the ...
16
votes
2answers
3k views

I've noticed some relationships with cosine and square root.

Yesterday I've noticed some relationships with cosine and square root. Anything interesting about it? I was trying to find the smallest width on an hexagon with radius $1.0$ and I noticed that I ...
15
votes
2answers
14k views

Square Fibonacci numbers

Are there Fibonacci numbers other than $F_0 = 0 = 0^2, F_1 = F_2 = 1 = 1^2,$ and $F_{12} = 144 = 12^2$ which are square numbers? If not, what is the proof?
15
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8answers
18k views

IMO 1988, problem 6

In 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. ...
15
votes
4answers
380 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 ...
14
votes
2answers
3k views

Are there infinitely many Mama's numbers and no Papa's numbers?

Just playing with square numbers, I made an interesting observation. $$11\times 11=121$$ $$12\times 12=144$$ $$13\times 13=169$$ $$.$$ $$.$$ $$.$$ $$20\times 20=400$$ $$21\times 21=441$$ $$.$$ $$.$$ $...
14
votes
2answers
772 views

If $u_1=1$ and $u_{n+1} = n+\sum_{k=1}^n u_k^2$, then $u_n$ is never a square.

Cute problem I saw on quora: If $$u_1 = 1,\qquad u_{n+1} = n+\sum_{k=1}^n u_k^2 $$ show that, for $n \ge 2$, $u_n$ is never a square. \begin{align} n&=1:& u_2 &= 1+1 = 2\\ n&=2:& ...
14
votes
1answer
4k views

Prove that $5$ is the only prime $p$ such that $3p + 1$ is a perfect square

Prove that $5$ is the only prime $p$ such that $3p + 1$ is a perfect square. I started off with assuming that $p$ is odd (since $2$ clearly does not satisfy). This would mean that $3p + 1$ is even. ...
14
votes
2answers
535 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 ...
13
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9answers
3k views

Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$

The formula basically is: The sum of all integers before and including $n$, plus all the integers up to and including $n-1$. This will find $n^2$. $$ \sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2 $$
13
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3answers
20k views

A number is a perfect square if and only if it has odd number of positive divisors

I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it. For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ ...
13
votes
2answers
1k views

Discrete math. Finding a perfect square.

The problem is: Find all natural numbers $n$ for which $2^n + 1$ is a perfect square? I am having a bit of trouble finding a generic way of finding these numbers. Of course the first obvious solution ...
13
votes
4answers
5k views

Could a square be a perfect number?

A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even number ...
13
votes
1answer
976 views

What finite groups always have a square root for each element?

If $G$ is an odd cyclic group of order $n$, then each element $g$ of $G$ has another element $h$ such that $h^2=g$. This is because $2 x = y \mod n$ is solvable for $x$. (Note this is not the same as ...
12
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6answers
3k views

Prime Numbers And Perfect Squares

Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square. One obvious solution is $p = q$ and under such a situation all primes $p$ and $q$ will satisfy. Further if $p \neq q$ ...
12
votes
2answers
2k 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?
12
votes
5answers
2k views

Why does this pattern in powers happen? [duplicate]

Hopefully this not to frivolous to ask, I have wondered it for years. There is a strange pattern in the differences between various power. For example if you square the numbers from one to ten the ...
12
votes
3answers
1k views

Can we prove a stronger claim?

Here it is asked whether for every $k>1$, there is a prime $p$ and $w_1,w_2,\cdots ,w_k>1$, such that $p$ divides $w_1^2+w_2^2+\cdots +w_k^2$, but none of the summands. Can we prove there is ...
12
votes
3answers
1k views

$(a+b)^2+4ab$ and $a^2+b^2$ are both squares

I cannot find a complete answer to the following problem (this is the source): Q. Find all positive integers $(a,b)$ for which $(a+b)^2+4ab$ and $a^2+b^2$ are both squares. Just something: ...
11
votes
4answers
426 views

Prove $n! +5$ is not a perfect square for $n\in\mathbb{N}$

I have a proof of this simple problem, but I feel that the last step is rather clunky: For $n=1,2,3,4$ we have $n!+5=6,7,11,29$ respectively, none of which are square. Now assume that $n\geq 5$, then:...
11
votes
3answers
254 views

Show that $a+b+c=0$ implies that $32(a^4+b^4+c^4)$ is a perfect square.

There are given integers $a, b, c$ satysfaying $a+b+c=0$. Show that $32(a^4+b^4+c^4)$ is a perfect square. EDIT: I found solution by symmetric polynomials, which is posted below.
11
votes
4answers
433 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 ...
11
votes
4answers
319 views

Prove or disprove that $8c+1$ is square number.

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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