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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17
votes
1answer
475 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 ...
11
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0answers
257 views

Is there $n \geq 2$ such that $1^1 + 2^2 + \dots + n^n$ is a perfect square?

Define $S_n = 1^1 + 2^2 + 3^3 + \dots + (n - 1)^{n - 1} + n^n$. It is clear that $S_1 = 1^1 = 1$ is a perfect square. However, I have found no others so far. Question: Does there exist $n \geq 2$ ...
11
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0answers
222 views

Are $(2,28)$ and $(5,3207)$ the only solutions $(m,n)\in\mathbb{N}^2$?

I noticed something as I was playing around with prime numbers. By denoting $p_i$ the $i^{\text{th}}$ prime number, I discovered the following: $$ \begin{align}\prod_{i=1}^2\left(p_i^{ \ 2}+i\right)&...
10
votes
1answer
236 views

Considering the equation, $6 + (2k+1)\sum_{n=1}^{2k+1}p_n^{ \ \ 3}(-1)^{n+1} = x^2$.

I noticed that, $$\begin{align}3(2^3 - 3^3 + 5^3) + 6 &= 18^2 \\ \text{and } \qquad 5(2^3 - 3^3 + 5^3 - 7^3 + 11^3) + 6 &= 74^2.\end{align}$$ These equations are of the form, $$6 + (2k+1)\sum_{...
10
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0answers
193 views

Can the sum of powers of the first primes be a square?

Let $p$ be a prime and $u\ge 1$ be a positive integer. Define $$\begin{align} S(p,u) &:= \sum_{q\text{ prime, }q \le p} q^u \\ &= 2^u+3^u+\cdots +p^u\end{align}$$ I wonder whether $S(p,u)$ ...
10
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1answer
158 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 N^+\...
9
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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 + ...
9
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0answers
256 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 ...
8
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0answers
1k views

Sum of three consecutive cubes equals a perfect square

I have found this problem in an old German textbook: Find all sets of three consecutive integers such that the sum of their cubes is a perfect square. We can write $$S = (x-1)^3 + x^3 + (x+1)^3 = (x-...
8
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0answers
381 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 ...
6
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0answers
226 views

Prove that every integer greater than 1 is sum of square and squarefree.

I tried to put some sieve method on this. Here, we denote the classic squarefree sieve function $\sum_{d^2|a}\mu(d)$ (notice that if $a$ is squarefree, the value is 1, and if not squarefree, the value ...
6
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0answers
592 views

Is there any perfect square in the sequence $12,123,1234,12345,…$?

Let $x_n$ be the number constructed by concatenating the first $n$ positive integers. Is there any perfect square in the sequence $(x_n)_{n ≥ 2}$ ? This is OEIS A007908. This is the sequence of ...
6
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0answers
283 views

When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ ...
5
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0answers
181 views

Prove that $\frac{m!}{200}\neq50x^2+51x+13$ and is my working correct?

Where $m$ and $x$ any real non-negative interger values. Prove that $$\frac{m!}{200}\neq50x^2+51x+13$$ Where $m!\geq20$ I understand that it may link into a trivial part of Brocard's problem. $$\frac{...
5
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0answers
137 views

Integer solutions for $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$

Consider the equation $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$. For $x \le 16$, the equation has the following integer solutions: $$ \begin{matrix} x = 0 & y = 0 \\ x = 1 & y = 0 \\ x = 4 &...
4
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0answers
34 views

is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere?

is there a function $\alpha(x)$ that follows these rules and is differentiable everywhere? just to keep track $\frac{a}{b}$ is the simplest form of $x$ when $x$ is rational. $\frac{c}{d}$ is the ...
4
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0answers
95 views

a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
4
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0answers
100 views

Is $p^2+q^2+r^2=3^k$ with primes $p,q,r$ solvable for every odd positive integer $k\ge 3\ $?

For the positive odd integers $3\le k\le 25$, the equation $$p^2+q^2+r^2=3^k$$ with primes $p,q,r$ is solvable. Here is one solution for every exponent , calculated with PARI/GP : ...
4
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0answers
100 views

Are there other values of $n$ that generate $p^2$?

I found a pattern that looks quite interesting. $$\begin{align} 2(4 + 2) + 13^3 &= 47^2 \\ 2(4 +5) + 7^3 &= 19^2 \\ 2(4 + 8) + 1^3 &= 5^2.\end{align}$$ It seems at first that if $p$ is ...
4
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1answer
72 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?
3
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0answers
51 views

Are there infinite square free integers $4x+1, 4x+2, 4x+3$

Clearly, $4$ divides $(x)(x+1)(x+2)(x+3)$ Likewise, if a prime $p^2$ divides $x$, $p^2$ does not divide $(x+p)(x+2p)\dots(x+[p-1]p)$. Does it follow that there are an infinite number of instances $...
3
votes
1answer
761 views

What is the difference between, a “square” and a “perfect-square”, number?

Is, "36", a perfect square? I know that, "4" is a perfect square. Similarly, "1","9","25", are "perfect-square"s.
3
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0answers
87 views

Prove that an odd square cannot be a pseudoprime with both base 2 and base 3

Background: The Baillie PSW primality test 1 tests if the number is a square before the Selfridge parameter selection. The Mathematica implementation of PrimeQ does ...
3
votes
0answers
232 views

Which natural numbers are the sum of three positive perfect squares?

In this question : Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $? I asked for the classification of the natural numbers being the sum of three ...
3
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0answers
144 views

Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $?

Let $S$ be the set of positive integers $n$, such that the equation $$a^2+b^2+c^2=n$$ has a solution in non-negative integers but not in positive integers. Since $n\in S$ if and only if $4n\in S$, we ...
3
votes
1answer
153 views

Factorials, squares and Bertrand's postulate

With Bertrand's postulate at hand, it is easy to see that $n!$ is never a square for $n\ge 2$ (because there is a prime between $n/2$ and $n$). But are there more elementary proofs of that fact?
3
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0answers
107 views

Determine when $(3^x - y^3)(x^3 - 3^y)$ is a perfect square

Determine all integers $x,y$ such that $f:=(3^x-y^3)(x^3-3^y)$ is a perfect square. What I have thought: the problem seems a bit hard to begin with. I first tried the situation that $\gcd(x,3)=\gcd(y,...
3
votes
1answer
81 views

Are all solutions of $ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$ periodic?

Let $n$ be a strict positive integer and $x$ is a complex number. Define $f_n(x)$ as one of the solutions to $$ f_n (x)^{2n} + f_n ' (x)^{2n} = 1$$ Where the derivative is with respect to $x$. Why ...
3
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0answers
285 views

Arithmetic progression of squarefree integers?

Let $x$ be a given positive integer. I'm intrested in the longest arithmetic progression of squarefree integers within the interval $(x,x^2)$. Both constructive and nonconstructive results. For ...
3
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0answers
49 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 $p_n \cdot p_{m+n} < p_{n+1}^2$?
3
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0answers
64 views

Formula for numbers whose iterated distance from next square has fixed point $2$

Consider a function $R: \mathbb N \to \mathbb N,$ $R(n)$ is the distance to the next square greater or equal to $n$. For example, $R(5)=4$ and $R(16)=0.$ Function $R$ has two fixed points, $0$ and $...
2
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1answer
65 views

$\frac{b^{2n}+b^{n+1}+3b-5}{b-1}$ is square

Find all $b>5$ so that $x_n = \frac{b^{2n}+b^{n+1}+3b-5}{b-1}$ is square for all sufficiently so large integers n. I think the only value of $b$ is 10. If there is $p \in \mathbb{P}$ (prime), $p \...
2
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0answers
92 views

How do I prove 8(x)! + (2x-1)² where x is an integer greater than or equal to 3, is never a perfect square.

$8(x)! + (2x-1)^2 = a^2$, so $8(x)! = a^2-(2x-1)^2$, for $x \geq 8$, $64|\text{LHS}$, but we can see 8 doesn't divide both of $(a-2x+1)(a+2x-1)$, so 64|one of them, so both one of them is $0 \pmod {64}...
2
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3answers
62 views

Finding the magic number as following

Let $s$ and $t$ be distinct positive integers with $s+t$ and $s-t$ are a square numbers. A pair $(s,t)$ called magic if there is exist positive integer $u$, such that $12s^2 + t^2 = 4t^2u^3$. Does it ...
2
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0answers
58 views

Proof of the remarkable formula for the n-th non-square?

The OEIS's A000037 entry makes the remarkable claim that every non-square number is given by the sequence $$a(n) = n + \Big\lfloor \frac 12 + \sqrt n\Big\rfloor$$ After looking through the entry, I ...
2
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0answers
96 views

If $n=N^2m$ for squarefree $m$, then $n$ is the sum of two squares if $m$ has no prime factor of the form $4k+3$

My question is ; Let $n$ be $n=N^{2} m$ , where m is a squarefree integer. Then $n$ can be written that as a sum of two integer squares, if $m$ contains no prime factor of the form $4k+3$. I have a ...
2
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0answers
60 views

An Algorithm For Finding Modular Square Roots

I've been working on this algorithm for awhile, and have made some progress, but have hit a stumbling block. For the cases I'm working on, square roots are guaranteed to exist (atleast trivial ones), ...
2
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0answers
16 views

Pairs of Squares with repunit difference — mistake by Selfridge and Lacampagne?

Dec. 1986: C.B. Lacampagne and J.L. Selfridge: Pairs of Squares with Consecutive Digits (Math. Mag. Vol. 59, no. 5: 270 – 275) www.jstor.org/stable/2689401) contains a list of pairs of square numbers ...
2
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0answers
63 views

An unusual pattern associated with perfect number

The pattern is that if you take a perfect number and place 5 at its unit digit. Then you have to square that number after squaring it the digits of the obtained number can be rearranged into two or ...
2
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0answers
39 views

When does a right triangle with integral perimeter and area have rational sides?

When does a right triangle with integral perimeter and area have rational sides? First find the sides a and b in terms of area $A$ and perimeter $P$. $A = ab/2 $, so $ab = 2A$. $\begin{array}{rl}\\...
2
votes
1answer
64 views

When does a certain number is a perfect square

I've the following number: $$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$ Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
2
votes
0answers
82 views

Finding squares that add to a certain sum

I have been looking for an algebraic way to solve for 2 squares when given a sum, but I got nothing so far. For example: $$y^2 + x^2 = 9797$$ I thought that the solution would have to do something ...
2
votes
1answer
58 views

For what values of $a$ does $b\in\mathbb{N}$?

I've the following function: $$b=\frac{1}{2}-\frac{1}{r-2}+\sqrt{\frac{1}{4}-\frac{r-3}{(r-2)^2}+\frac{a(1+a)}{r-2}+\frac{a(a^2-1)}{3}}$$ I know that: $\text{a}\ge1$ and $a\in\mathbb{N}$; $\text{r}\...
2
votes
2answers
2k views

Square root of zero

I'm old 35 but starting just now with maths, so sorry if I ask non complex questions. 0 is the only number that just has one square root. Is the explanations for this simply that 0 in arithmetic does ...
2
votes
0answers
83 views

When can we factor $N$ efficiently with a representation $N^2=a^2+b^2$?

Here : Can the sum of two squares be used to factor large numbers? an idea to factor a large number $N=a^2+b^2$ is shown. Under which conditions does a representation $$N^2=u^2+v^2$$ exist , such ...
2
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0answers
1k views

Are there 3 definitions of rectangular numbers?

I'm a little bit confused about the definition on rectangular numbers, which is also giving me doubts about its function (or idea). So there's the definition of that all rectangular can be found with ...
2
votes
0answers
141 views

Limits on difference between a perfect square and a perfect cube?

Are there known bounds on these differences? $378661^2-5234^3=17$ is the best I found for small numbers, which suggests to me that there's maybe a lower bound that goes as the cube root of the number ...
2
votes
0answers
57 views

On numbers of the form $\pm x^2\pm N_k$, with $x\geq 1$ integer and $N_k$ denoting primorials

In this post we consider positive integers $a$ of the form $$\pm x^2\pm N_k\tag{1}$$ with $x\geq 1$ integer and for integers $k\geq 1$ denoting $$N_k=\prod_{j=1}^k p_j$$ as the primorial of order $k$,...
2
votes
0answers
105 views

Sums of descending squares

I am interested in integers that can be expressed as a sum of squares. Specifically I am interested in integers that can be expressed as follows: $n=6*Sum (k^2+(k-a)^2+(k-2a)^2.....1^2)$ These ...
2
votes
0answers
482 views

Polynomial that is a square in every positive integer

Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a polynomial with integer coefficients such that $p(n)$ is a perfect square for every $n\in\mathbb{N}^{*}$. Is it true that exists a polynomial $q$ with ...