for all integers a. This result gives us the rudimentary Fermat Compositeness Test: If a 6≡ a (mod n) for some integer a, then n is composite. While this has the advantage of being computationally simple, it has the distinct disadvantage of failing for some composite n and choice of a. Take, for example, n = 341 = 31 · 11 and a = 2. A quick computation tells us that 2 ≡ 2 (mod 341). Fortunately...

We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...

The sensitivity of trajectories over finite-time intervals t to perturbations of the initial conditions can be associated with a finite-time Lyapunov exponent lambda, obtained from the elements M(ij) of the stability matrix M. For globally chaotic dynamics, lambda tends to a unique value (the usual Lyapunov exponent lambda(infinity)) for almost all trajectories as t is sent to infinity, but for...

The spatially varying tensor fields that arise in magnetic resonance imaging are difficult to visualize due to the multivariate nature of the data. To improve the understanding of myocardial structure and function a family of objects called glyphs, derived from superquadric parametric functions, are used to create informative and intuitive visualizations of the tensor fields. The superquadric g...