For any positive integer n, define an iterated function $f(n)=left{ egin{array}{ll} n/2, & mbox{ $n$ even, } 3n+1, & mbox{ $n$ odd. } end{array} ight.$ Suppose k (if it exists) is the lowest number such that fk(n)

We discuss the concept of coding over the ring of integers modulo m. This method of coding finds its origin in the early work by Varshamov and Tenengolz. We first give a definition of the codes followed by some general properties. We derive specific code constructions and show computer-search results. We conclude with applications in 8-phase modulation and peak-shift correction in magnetic recording systems.

A speedup of Lenstra's Elliptic Curve Method of factorization is presented. The speedup works for integers of the form N = PQ2, where P is a prime sufficiently smaller than Q. The result is of interest to cryptographers, since integers with secret factorization of this form are being used in digital signatures. The algorithm makes use of what we call Jacobi signatures. We believe these to be of independent interest.