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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4answers
394 views

Is this relationship already known?

I like math because it's a puzzle to me, but am really not very good at it. But I figured out the relationship below myself. Just curious, is this already pretty common knowledge? Kind of proud of ...
17
votes
3answers
895 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) = ...
0
votes
1answer
20 views

Sum of the squares of $5$ consecutive positive numbers can not be a perfect square.

Proof that, Sum of the squares of $5$ consecutive positive numbers can not be a perfect square. As far I did, $(n-2)^2 + (n-1)^2+n^2 + (n+1^2) + (n+2)^2$ $=2(n^2+4) + 2(n^2 + 1) + n^2$ $=5(n^2 + ...
3
votes
2answers
54 views

Prove $|x|^2$ = $x^2$

My first attempt at this proof divided into 2 cases, one where $x^2$ is greater than or equal to 0, and another where $x^2$ is less than 0. For the first case, I said that the definition of absolute ...
2
votes
2answers
59 views

A triple of pythagorean triples with an extra property

I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align} (x-z)^2+y^2 &\text{ is a perfect square,}\\ x^2+y^2 &\text{ is a perfect ...
0
votes
3answers
29 views

Squaring a binominal

I have a really simple question that I can't find the answer to. In a algebra test, I was asked to simplify $(5 + $$\sqrt{3}$)$^2$. What I did was to square each term individually: ...
0
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0answers
45 views

Solving for 2 unknowns in a perfect square

Basically was looking at a sequence of perfect squares for a given constant integer A, where in the first instant we can easily and trivially generate a sequence of perfect squares using the ...
-1
votes
3answers
72 views

How to approach solving for $x$? $(1+x)^2 = 1.21$

I'm trying to solve for $x$ in the following equation, but don't know how to solve it without using a graphing calculator. $$(1+x)^2 = 1.21$$ Is there a rule I'm supposed to follow?
2
votes
1answer
60 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
2
votes
2answers
35 views

Perfect square is $0$ or $1$ modulo $4$ . [closed]

Prove that for every integer $n$ either $n^2 \equiv 0\pmod{4}$ or $n^2\equiv 1\pmod{4}$
5
votes
0answers
39 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 ...
1
vote
4answers
43 views

Does every linear integer polynomial give a square at some integer?

My question is, if you have some function $$f(x)=nx+c$$ which accepts only integer inputs of $x$, where $n>0$ and $c$ are fixed integer constants, can you always find an $x$ such that $$f(x)=k^2$$ ...
0
votes
1answer
26 views

How to find product of factors of a number if the factors are perfect square?

Let M be the set of all the distinct factors of the number N = 6^5 * 5^2 * 10, which are perfect squares. Find the product of the elements contained in the set M. N = 2^6 * 3^5 * 5^3 Even power of ...
0
votes
1answer
37 views

More efficient method of computing the square root of $-1 \mod p$

I am currently doing collecting some preliminary data about elliptic curves over finite fields of order $p$ where $p$ is a prime congruent to 1 mod 4. Part of the data collection process requires me ...
5
votes
3answers
91 views

If $x-y = 5y^2 - 4x^2$, prove that $x-y$ is perfect square

Firstly, merry christmas! I've got stuck at a problem. If x, y are nonzero natural numbers with $x>y$ such that $$x-y = 5y^2 - 4x^2,$$ prove that $x - y$ is perfect square. What I've ...
1
vote
2answers
36 views

A Game on Triangular and Square Numbers

A game is played by $n$ people $A_1, A_2,...,A_n$ with the following rules: 1. The $n$ people take turn to call out a positive number, in ascending order, $\quad$i.e. $A_1$ says 1, $A_2$ says 2, ...
1
vote
3answers
49 views

Number of values in square root in different cases

I have two equations: $x = \sqrt{16}$ $x^2 = 16$ In first case I think there will be two value of $x = \pm4$. Because $(-4) \cdot (-4) =( +4) \cdot(+4) = 16$ In the second case I am confused. It ...
-1
votes
2answers
68 views

Is it true that if any two of $m$, $n$, and $mn$ are sums of two integer squares, then so is the third? [closed]

Where m and n are positive integers. Prove or give a counter example.
0
votes
1answer
52 views

My lucky number

I don't know should I ask in Puzzling or not. Sorry for that if I'm wrong! And my question is, I have a lucky number that has four digits which the first three numbers and the last three numbers are ...
4
votes
3answers
45 views

There is no square number that is $3 \mod 4$

Prove that the following cannot be true: There is no square number that is $3 \mod 4$ $x^2 \equiv 3 \mod 4$, I started with examples: $1^2 \mod 4=1$ $2^2 \mod 4=0$ $3^2 \mod 4=1$ I am more ...
3
votes
2answers
45 views

Dividing the squares $1^2,2^2,\ldots,54^2$ into three equal groups with the same total sum

Is it possible to divide the squares $1^2,2^2,\ldots,54^2$ into three groups, each of which contains $18$ squares, such that the sum of squares within each group is the same for all three groups?
1
vote
0answers
29 views

Impossible form of a triangular number

Show that there are no positive integers $t,i,j$ with $j>i$ such that: $\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$ If possible provide an elementary proof. I believe the statement is ...
12
votes
3answers
284 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: ...
1
vote
3answers
46 views

$Dm^2 - n^2D^2$ is a perfect square then $D$ is the sum of two squares

How do I show that if $$Dm^2 - n^2D^2$$ is a perfect square for some integers $m$ and $n$ ($n \neq 0$), $D$ is the sum of two (non-zero) perfect squares? I tried solving for $D$ but that only gives me ...
1
vote
1answer
75 views

Find all positive integers $n$ such that $2^8+2^{12}+2^n$ is a perfect square

Find all positive integers $n$ such that $2^8+2^{12}+2^n$ is a perfect square. For $n=2$ and $n=11$, $2^8+2^{12}+2^n$ is a perfect square. How to find a closed form?
1
vote
3answers
89 views

The sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$ [duplicate]

Prove that the sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$. Can someone pls help and provide a solution for this and if possible explain the question
1
vote
1answer
54 views

When are both $x$ and $ y$ rational in $ x^2 + y^2 = k$

Under what conditions for k such that the circle equation $x^2 + y^2 = k$ has rational solutions for both $x$ and $y$? For example, when $k = 4$, $\{x=2, y=0\}$ is a set of rational solutions. But ...
0
votes
3answers
43 views

Show that solutions of $a^3+b^3=c^3+d^3$ and $a+b=c+d$ for distinct $a,b,c,d$ are non-existent.

Question: Show that there do not exist real, distinct $a,b,c,d$ such that $a^3+b^3=c^3+d^3$ and $a+b=c+d$. My attempt: $$a^3+b^3=c^3+d^3\implies(a+b)(a^2+b^2-ab)=(c+d)(c^2+d^2-cd)$$ Since ...
0
votes
0answers
16 views

Understanding Ferrari's Solution

I'm trying to understand how Ferrari's Solution works. Thanks to this post I understand that we are solving this for $y$ to find the perfect square: $$y^3 + \frac{5\alpha y^2}{2} + (2\alpha^2 - ...
0
votes
0answers
24 views

Show that there exists infinitely many $n$ such that $u_n$ is not a perfect square

Let $a, b, c$ be positive integers. Define the sequence $u_n$: $$u_n = a \cdot 2^n + b \cdot 5^n + c$$ Show that there exists infinitely many $n$ such that $u_n$ is not a perfect square. Source: ...
5
votes
1answer
86 views

Product of numbers $\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$ is a perfect square

Let $A$ be the product of $2^{100}$ numbers of the form $$\pm \sqrt{1} \pm \sqrt{2} \pm \dots \pm \sqrt{100}$$ Show that $A$ is an integer, and moreover, a perfect square. I found a similar problem ...
2
votes
1answer
71 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 ...
0
votes
0answers
20 views

Square Root of 1.4 using series expansion

does anyone have a formula or knows how to do a series expansion of a square root? I have tried researching it, and looking through formulas but can only find formulas that work for Sine and Cosine ...
7
votes
1answer
187 views

Binomial coefficients that sum to a perfect square

I was wondering about a property of a sum I recently saw. $${8\choose2}+{9\choose 2}+{15\choose2} + {16\choose2}=17^2$$ And if we increment the terms $${9\choose3}+{10\choose 3}+{16\choose3} + ...
3
votes
0answers
38 views

$n$ integers, $a_i a_j +1$ all perfect squares [closed]

Find all numbers $n$ so that there exists $n$ integers $a_1, a_2, ..., a_n$: $a_i \ge 2$ and $a_i\cdot a_j +1 (\forall i\not = j)$ are all perfect squares.
0
votes
0answers
24 views

If a, b, and k are positive integers such that k = (a² + b²)/(ab + 1), show that k is a square. [duplicate]

If a, b, and k are positive integers such that k = (a² + b²)/(ab + 1), show that k is a square. This is the work I have done so far: k = (a² + b²)/(ab + 1) abk + k = a² + b² a² - abk = k - b² ...
1
vote
1answer
52 views

Proof by cases: If n is a positive integer and there is a perfect square in $\{k \in \mathbb{Z} | n \leq k \leq 2n\}$

If n is a positive integer and there is a perfect square in $\{k \in \mathbb{Z} | n \leq k \leq 2n\}$, then there is a perfect square in $\{k \in \mathbb{Z} | n + 1 \leq k \leq 2n + 2\}$. There seems ...
2
votes
1answer
195 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 ...
0
votes
2answers
34 views

If a number cannot be…

If there exists such a number which cannot be divided by some other number, which is equivalent to, or smaller than the square root of itself, it is a prime number. This is a rather trivial theorem ...
53
votes
7answers
1k 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 $k$ ...
1
vote
1answer
64 views

There exists a perfect square between $n$ and $2n$ [duplicate]

I need help proving the following mathematical statement: Prove that for every $n$ there is a $k$ such that $n \leq k^2 \leq 2n$ where $n,k\in \mathbb{N}$. Could someone get me started or give ...
0
votes
2answers
74 views

Can a square number be expressed as sum of squares of two other members.? [closed]

Is there any theorem to tell if square of a number can be expressed as sum of squares of two other distinct numbers. I have one such set. {5 4 3} 5^2 = 4^2 + 3^2 Given a number n how to find if ...
0
votes
3answers
72 views

Explicit formula for a recurrance relationship $A_n = A_{n-1} + 2n + 1$

$$a_n = a_{n-1} + 2n + 1 $$ $$ a_0 = 1 $$ $$ a_1 = 4 $$ $$ a_2 = 9 $$ I know the basics of how to use characteristic polynomials, but I'm not sure how the $2n$ would be represented in the ...
1
vote
1answer
70 views

Is it possible to subtract a perfect square from another number to make it a perfect square?

A number $c$ is given. We need to find a number $0<k<c$ such that $c^2 - k^2$ is a perfect square. (if it is possible) $c$ and $k$ can be any positive integer. What I tried is- I iterated ...
0
votes
3answers
32 views

Sum of Odd Numbers make Squares

Look at this: 1 (+3) 4 (+5) 9 (+7) 16 (+9) 25 (+11) 36 (+13) 49 And so forth, you get the idea. Why do they make up this pattern? And is there any special name for this ...
2
votes
2answers
77 views

The amount of times a number need to be squared rooted

Consider the following recursive function $f()$ def f(x,n=0): if x<2: return n return f(math.sqrt(x),n+1) $f(x)$ calculates the number of ...
0
votes
5answers
102 views

Prove: If $n^2$ is odd, then $n$ is odd. [duplicate]

$n$ is a natural number. I want to prove that, if the square of $n$ is odd, then $n$ itself is odd. Any hints welcome and preferred. Thank you!
1
vote
2answers
58 views

If the square of a natural number is odd then this number is odd.

My book says that We represent $n$ as $n=2k+1$ where $k$ is from natural numbers or $k=0$ Then $n^2=(2k+1)^2=4k^2+4k+1$. We write $n^2=2(2k^2+2k)+1$ where $2k^2+2k$ is natural number or ...
1
vote
1answer
49 views

Are Fibonacci numbers with a square prime index always divisible by $F_p$?

I am doing some research on sequences and I need some help. The sequence of $F_{p^2}$ seems sort of different. It seems that because the index only has one distinct prime factor, as a result the only ...
3
votes
3answers
42 views

Given area of square $= 9+6\sqrt{2}$ Without calculator show its length in form of $(\sqrt{ c}+\sqrt{ d})$

$\sqrt{9+6\sqrt{2}}$ to find length But how do I express the above in the form of $\sqrt{c} + \sqrt{d}$.