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 […]

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.

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 […]

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?

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 + […]

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 […]

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 […]

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 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 […]

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 $k-1$ […]

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