An error-correcting code can be defined as a set of functions mapping $P$, called the set of places, to $A$, called the alphabet. With classical Generalized Reed-Muller Codes, $P$ is an $m$-dimensional vector space $F^m$ over a finite field $F$, and $A$ is just the finite field $F$. Then, $C= \text{ GRM}(\nu,m)$ is defined to be the set of all functions from $P$ to $A$ which, when represented as a polynomial of minimal degree through Lagrange interpolation, (see [2], for example) has degree less than or equal to $\nu$.

This procedure can be generalized. $C=A_{\nu}$ is taken to be an element of the filtration of some filtered $F$-algebra $B$. $A$ is another $F$-algebra, and $P = \text{HOM}_{ALG}\ (B,A)$. Then, $C=C_{\nu}(B,A)$ is the set of elements of $B_{\nu}$ viewed as functions from $P$ to $A$ via $b(x) := x(b)$ for $x\in P$ and $b\in B$.

In this note we give a construction of a Pythagorean ring $$P_n = \{ \ \lt x,y,z > \in {\mathbb Z} ^3\ :\ x^2 + y^2 = z^2,\ z-y = n \ \} ,$$ where $n$ is a fixed integer.

The Twin Prime Conjecture states that the number of twin primes is infinite. Many attempts to prove or disprove the conjecture have failed. The objective of this note is to tie the Twin Prime Conjecture to complex variable theory and prove some results that make it possible to consider the conjecture from a complex variable viewpoint rather than from a purely number theoretic one.

A topology $\tau$ on a non-empty set $X$ is called a complementary topology if each open set $U$ in $\tau$, its complement $X - U$ is also in $\tau$. These topologies, maximal ideals and their relation to ultrafilters were characterized by this author. In this paper, the structure of filters in $\tau$ are investigated. Finally, the ultrafilters are characterized.