Is there a 'simple' 2 dimensional cellular automaton to generate all prime twins ? With 'simple' I mean not too many states per cell and not so many rules. Thus a universal turing machine equivalent cellular automaton is not 'simple'.

What can I say about $x^4 \equiv -4 \mod p$ where $p$ is prime? In general what can I do with powers that are greater than $2$ and where I cannot use reciprocity, legendre/jacobi etc... In general what can I say about a quadratic polynomial modulo $p$: For instance $(x-1)^2 \equiv 1 \mod p$ By '...

I would like to learn the following: a) Prove that the equation $1 + x + x^2 = py$ has integer solutions for infinitely many primes $p$. b) Twin primes are those difference by 2. Show that 5 is the only prime belonging to two such pairs. Show also that there is a one-to-one correspondence betwe...

I need a justification for my observation. In general, we can list twin prime pairs in $(6n-1, 6n+1)$, where $n$ is some positive number. Of course, this is valid except $(3, 5)$. Now, I construct, for any such twin primes pair will satisfy the following my observation. $$4(6n-2)! = -3(1+2n)\pm...

I am so exited to learn math from this site. I posted the question today and I got good replies from members today itself. I will try to answer other number Theory questions in near future. With same confidence and motivation, I am sending TWO more questions to the members. These are also my obse...

Let $p=p(x_1,...,x_N)$ be a non-zero polynomial in $N$ variables (real coefficients). Let $\mathscr{S}$ be the Schwartz space on $\mathbb{R}^N$ and let $\mathscr{S}'$ be its topological dual (i.e. the space of tempered distributions). I know that the map $\mathscr{S}'\to \mathscr{S}', \ \ \ T\map...