Articles of prime factorization

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 […]

A fast factorization method for Mersenne numbers

Given a prime number $p$ and a Mersenne number $M=2^p-1$: Is it true for every prime factor $q$ of $M$ that $q\equiv1\pmod{p}$? For example, $p=29$ and $M=536870911=233\cdot1103\cdot2089$: $ 233=29\cdot 8+1$ $1103=29\cdot38+1$ $2089=29\cdot72+1$ If yes, then is there a method which exploits this fact in order to factorize $M$ (or decide it is prime), which is […]

Factorials and Prime Factors

I need to write a program to input a number and output it’s factorial in the form: $4!=(2^3)(3^1)$ $5!=(2^3)(3^1)(5^1)$ I’m now having trouble trying to figure out how could I take a number and get the factorial in this format without actually calculating the factorial. Say given 5 need to get result of $(2^3)(3^1)(5^1)$ without […]

Number Theory Prime Reciprocals never an integer

I’m in number theory and I currently have these problems assigned as homework. I’ve looked through the sections containing these problems and I’ve solved/proved most of the other problems, but I can’t figure these ones out. For $n>1$, show that every prime divisor of $n!+1$ is an odd integer that is greater than $n$. Assuming […]

Proof: no fractions that can't be written in lowest term with Well Ordering Principle

My question is the exact same question as the one in this post but I commented on it but it’s from a year ago so I just wanted to bump it and see if I could get a response: Prove that there's no fractions that can't be written in lowest term with Well Ordering Principle […]

Factor $10^n – 1$

How can the prime factors of $10^n – 1$ be found? $9 = 3^2$ is obviously a factor. If $n = p-1$, $p$ is a factor from Fermat’s Little Theorem. I am stuck beyond that.

Are there always at least $3$ integers $x$ where $an < x \le an+n$ and $\gcd(x,\frac{n}{4}\#)=1$

The answer seems to be yes. Please let me know if I have made a mistake in my argument or if there is an opportunity to make the argument simpler, clearer, or more standard. Let $a \ge 1, n \ge 2$ be integers. Let $\frac{n}{4}\#$ be the primorial for $\left\lfloor\frac{n}{4}\right\rfloor$ Let $\gcd(a,b)$ be the greatest […]

Proof that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too

I’m trying to figure out how to proof that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too. Can someone help me please?

Counting $x$ where $an < x \le (an+n)$ and lpf($x$) $ \ge \frac{n}{4}$ and $1 \le a \le n$

Let lpf($x)$ be the least prime factor of $x$. It seems to me that if: $1 \le a \le n$ $n \ge 128$ $an < x \le (an+n)$ lpf($x$) $\ge \frac{n}{4}$ Then, for all $y$ where: $an < y \le (an+n)$ $ y \ne x$ It follows that lpf($y$) $< \frac{n}{4}$ Here’s my reasoning: Claim […]

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?