Articles of algebra precalculus

How does one prove that $(2\uparrow\uparrow16)+1$ is composite?

Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: $$F_1=2+1\to prime$$ $$F_2=2^2+1\to prime$$ $$F_3=2^{2^2}+1\to prime$$ $$F_4=2^{2^{2^{2}}}+1\to ?$$ And so on. Amazingly, it has been found that $F_1$ through $F_{15}$ to […]

If both integers $x$ and $y$ can be represented as $a^2 + b^2 + 4ab$, prove that $xy$ can also be represented like this …

There is a set $Q$ which contains all integral values that can be represented by $$a^2 + b^2 + 4ab$$, where $a$ and $b$ are also integers. If some integers $x$ and $y$ exist in this set, prove that $xy$ does too. I really have no idea how I can go about solving this. I […]

Find the $least$ number $N$ such that $N=a^{a+2b} = b^{b+2a}, a \neq b$.

When I graphed the relation $a^{a+2b}=b^{b+2a}$ , it gives a graph similar to $y=x$. However, the question explicitly states that $a \neq b$. So does that mean that no such $N$ exists ? What happens when the problem is generalized as $N=a^{ma+nb}=b^{mb+na} $ ? Can anybody help as to what should be done ? Thanks […]

derivation of simple linear regression parameters

I know there are some proof in the internet, but I attempted to proove the formulas for the intercept and the slope in simple linear regression using Least squares, some algebra, and partial derivatives (although I might want to do it wituout partials if it’s easier). I’ve posted my attempt below. I don’t know what […]

How to find $n$'th term of the sequence $3, 7, 12, 18, 25, \ldots$?

$$3, 7, 12, 18, 25, \ldots$$ This sequence appears in my son’s math homework. The question is to find the $n$’th term. What is the formula and how do you derive it?

How do I isolate $y$ in $y = 4y + 9$?

$y = 4y + 9$ How do I isolate y? Can I do $y = 4y + 9$ $\frac{y}{4y} = 9$ etc Also some other questions please: $\frac{5x + 1}{3} – 4 = 5 – 7x$ In the above (1), if I want to remove the ‘3’ from the denominator of the LHS, do I […]

Solve $\epsilon x^3-x+1=0$

I’m trying to find the expansion for the roots of this equation. I’ve found one root as $x\sim 1+\epsilon $. Now considering the dominant balance I want to rescale so that $\epsilon x^3\sim O(x) \Rightarrow x=O(1/\sqrt\epsilon )$ Setting $x=y(1/\sqrt\epsilon )$ where $y=O(1)$ I get the new equation $$y^3-y+\sqrt\epsilon=0$$ Now I want to substitute in $y\sim […]

How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one?

I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?

How to solve the given problem of simple interest?

The problem statement is: What annual instalment will discharge a debt of 1092 due in 3 years at 12% simple interest? Now, what I know is Simple interest =( principal* Rate per annum*Time in years)/(100) Here, R= 12℅ ,T=3 years but I don’t understand how to move forward.I am not getting the meaning of the […]

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that […]