Consider the Eisenstein integers. Let p be an odd prime. $$x^p+y^p=z^p$$ with $x,y,z$ in the Eisenstein integers, excluding solutions in $\mathbb{Z}$ or $w\mathbb{Z}$ for a unit $w$. Are there any nontrivial solutions ? By trivial I mean excluding units and zero's for $x,y,z$. What has been found...

An Elementary Proof of the Twin Prime Conjecture The set of prime number can be written: $$\bigcap_{c \notin \Bbb{P}} \Bbb{Z}\setminus c\Bbb{Z} = \Bbb{P} \tag{0}$$ We start with an elementwise addition that commutes with set complement and arbitrary intersection. For any family of sets $A_i \sub...

Here is a proof. Based upon all the operations used within it, I would like to know how to generalize it to other rings than $\Bbb{Z}$. Let $\Bbb{P}$ denote the set of $\pm$ the prime numbers but including $\pm1$. Define the set of composite numbers to be $C = \Bbb{Z} \setminus \Bbb{P}$. The set...

It is easy to square a number. So instead of computing $\exp(z)$ you can compute $\exp(z/2^k)^{2^k}$ for some suitably large positive integer $k$. This is a very simple way of accelerating the convergence of the Taylor series for $\exp$, also known as "argument reduction". The double-exponential ...

Consider the following procedure with four six-sided dice: Roll all four of them. If any of them result in a one, then re-roll those dice until they no longer are ones. Take the highest three numbers of the dice and sum them. Suppose we repeat this procedure six times and generate numbers $Y_1,...

Underwood Dudley published a book called mathematical cranks that talks about faux proofs throughout history. While it seems to be mostly for entertainment than anything else, I feel it has become more relevant in modern mathematics. Especially with the advent of arXiv, you can obtain research pa...