# 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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### $\frac{1}{5}\big((4 + \sqrt{15})^{2n} + (4 - \sqrt{15})^{2n} + 8\big)$ is the sum of 3 consecutive squares

Prove that for every positive integer $n$, the number $$\large \frac{(4 + \sqrt{15})^{2n} + (4 - \sqrt{15})^{2n} + 8}{5}$$ can be expressed as a sum of squares of three consecutive integers. ...
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### 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 ...
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### When is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square for $x\ge2$ and $x\in\mathbb{N}$

I've the following number: $$81 + 60 x (1 + x) (-2 + 5 x)$$ For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $81 + 60 x (1 + x) (-2 + 5 x)$ a perfect square?
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### For what value of $x$ is the following number a perfect square

I've the following number: $$1+12x^2(1+x)$$ For what value of $x\ge2$ and $x\in\mathbb{N}$ is the number $1+12x^2(1+x)$ a perfect square?
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### Number of spread numbers in the field $\mathbb{F}_{p}$

I'm reading a book defining a "spread number" in a prime field a number $s \in \mathbb{F}_{p}$ verifying $s(1-s)$ being a square number. Then, the book says without proof that: if $p = 1 (\mod 4)$ ...
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### Perfect squares relantionship [closed]

Find all natural numbers $\overline{xyzt}=10^3x+10^2y+10z+t$ who satisfies the following condition $$\sqrt{x}+2\sqrt{y}+3\sqrt{\overline{zy}}+4\sqrt{z}=t^2.$$
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### Using the 'reductio ad absurdum' to prove the non-existence of two positive integers given a condition

I found this problem in the 'logic' section of my discrete mathematics textbook (I am currently a freshman majoring in Computer Science). It is a demonstration problem and I feel like I might have ...
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### 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 ...
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### Numeric functions

Let $n$ and $k$ be positive integers. A function $f$:{$1, 2, 3, 4, ... kn$} --> {$1, ... 5$} is said to be good if $f (j + k) - f (j)$ is a multiple of $k$ for all $j = 1.2, ..., kn - k$ a) Prove ...
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### How can be proved $\sqrt{n^8+2\cdot 7^{m}+4}$ is irrational?

How can be proved $\sqrt{n^8+2\cdot 7^{m}+4}$ is irrational, where $n,m \in \mathbb{N}$. I tried to write: $$a^2-n^8=2(7^m+2)$$ and to try finding the last digit in the left and in the right side. ...
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### Pairs of perfect squares

Two perfect squares are said to be friendly if one is obtained from the other by adding the digit 1 on the left. For example, $1225 = 35 ^ 2$ and $225 = 15 ^ 2$ are friendly. Prove that there are in ...
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### How do I solve this equation for integer solutions? [closed]

How do I find integer solutions for the following equation? $x^2 + 2y = 1$
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### Pattern in the list of square-triangular numbers.

I was just reading about Square Triangular numbers on its Wikipedia page (https://en.wikipedia.org/wiki/Square_triangular_number) and noticed a peculiar pattern in them. Given below is the list of ...
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### A Diophantine Equation related to arithmetic progression: $T_n=a^n+b^n+c^n$.

I'm working on a problem that for complex $a,b,c$, when will these four numbers $a+b+c, a^2+b^2+c^2, a^3+b^3+c^3, a^4+b^4+c^4$ becomes an arithmetic progression with integer values. If we let this ...
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### why value of function with sqrt changes

I have this function (1) $f(x) = {\sqrt (x+1) - \sqrt x}$ then I make it (by multiply with : $\frac{\sqrt (x+1)+\sqrt x}{\sqrt (x+1)+\sqrt x}$) into: (2) $f(x) = \frac{1}{\sqrt (x+1)+\sqrt x}$ the ...
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### If $T_n$ is the $n$-th triangular number, there are an infinite number of $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2$ for all $n$.

If $T_n$ is the $n$-th triangular number, show that there are an infinite number of positive integers $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2$ for all $n$. This is inspired by an article in ...
I am trying to prove the following statement. Let $d = \gcd(x_1,x_2)$. If $x_1 x_2$ is a square, then ($x_1 = d M^{2}$ and $x_2 = dN^2$) or ($x_1 = -d M^{2}$ and $x_2 = -dN^2$), where $\gcd(N,M) =1$. ...