Articles of number theory

What would a base $\pi$ number system look like?

Imagine if we used a base $\pi$ number system, what would it look like? Wouldn’t it make certain problems more intuitive (eg: area and volume calculations simpler in some way)? This may seem like a stupid question but I do not remember this concept ever being explored in my Engineering degree. Surely there is some […]

Divisors of $2^kp^r$

Let $p$ be an odd prime number. What is the necessary and sufficient condition (in terms of $p$ and $k,r$) such that we can partition the divisors of $2^kp^r$ into two set with equal sum. You may want to LOOK HERE where the special case $2^kp$ is dealed.

Ideal class “group” of Lipschitz (fully-integer) quaternions

Let $H = \left\{a+bi+cj+dk \in \mathbb{H} : a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}$ be Hurwitz (or semi-integer) quaternions. Then $H$ is Euclidean, thus principal ideal domain. Now let $L = \left\{a+bi+cj+dk \in \mathbb{H} : a,b,c,d \in \mathbb{Z}\right\}$ be Lipschitz (or fully-integer) quaternions. It is easy to see that the right […]

Minimum amount of primes between squared primes

Conjecture: $\forall $ $p_{n}$, $p_{n+1} \in \mathbb{P}$, $\:$ $p^2_{n+1} = p^2_{n} +\omega_{n} p_{n} + \phi_{n} : \phi_{n} , \omega_{n} \in \mathbb{N} $ and $ \phi_{n} < p_{n}$, $\:$ $\Rightarrow$ $\:$$\exists$ atleast $ \omega_{n}$ prime numbers in the interval $(p^2_{n},p^2_{n+1})$ How would you prove this?

How to prove sum of two numbers of the two following forms can be equals to sum of two numbers not of the forms?

The two forms are: $\ 3x^2 + (6y-3)x – y\ $ $\ 3x^2 + (6y-3)x + y – 1, \ \ x,y \in \mathbb{Z}^{+}$ For example: $\ \ \ 5 = \ 3*1^2 + (6*1-3)*1 – 1\ $ ,when $\ x = y = 1\ $,of the two forms $\ 13 = \ 3*1^2 + […]

Primitive Pythagorean triple generator

I was wondering how to prove the following fact about primitive Pythagorean triples: Let $(z,u,w)$ be a primitive Pythagorean triple. Then there exist relatively prime positive integers $a,b$ of different parity such that $$z = a^2-b^2, \quad u = 2ab, \quad \text{and} \quad w = a^2+b^2.$$ I see how $z,u,w$ must form the sides of […]

The primes $s$ of the form $6m+1$ and the greatest common divisor of $2s(s-1)$

I had trouble coming up with a good title. Let $\psi(s) = 2s(s-1)$. I write $(\psi(s),\psi(s+2))$ to be the greatest common divisor of $\psi(s)$ and $\psi(s+2)$. Then if $s$ is prime and $(\psi(s),\psi(s+2))=12$ we have that $s=6m+1$ for some positive integer $m$. Conversely if $s=6m+1$ is prime then $(\psi(s),\psi(s+2))=12$. I would like to solve this […]

A question on primes and equal products

Is the following statement true; “For any odd prime number $p$ , any set of $p-1$ consecutive integers can not be partitioned into two subsets such that the elements of the two sets have equal product.” ( I think I read somewhere that Erdos proved it but now I can not remember where I read […]

How many ways are there of coloring the vertices of a regular $n$-gon

How many ways are there of coloring the vertices of a regular $n$-gon with all $p$ colors ($n,p \ge 2$), such that each vertex is given one color, and every color isn’t used for two adjacent vertices? If in a way to color not necessarily use all $p$ colors, then the answer is $a_n=(-1)^n(p-1)+(p-1)^n\;.$ If […]

Prove that if $a_0\geq 2$, $f(n)$ is not prime for some integer $z$.

Let $f(x) = a_kx^k + a_{k-1}x^{k-1} + \cdots + a_0$ be a polynomial with integer coefficients where $a_k \neq 0$. Prove that if $a_0 \geq 2$, $f(n)$ is not prime for some integer $z$. I know I will eventually have to show that a polynomial of degree $k-1$ with real coefficients has at most $k-1$ […]