The GL

After defining a meanfield by arithmetic means, using multiplicative characters of finite fields, its Potts Hamiltonian is exactly computed. Moreover, it proves to be invariant with respect to every change of basis in Fq over the prime field Fp

Constructing irreducible polynomials with prescribed level curves over finite fields

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomial P(X,Y)∈GF(q)[X,Y] with coefficients in the finite field GF(q) with q elements, with prescribed level curves Xc:={(x,y)∈GF(q)2|P(x,y)=c}

On the ranges of discrete exponentials

Let a>1 be a fixed integer. We prove that there is no first-order formula ϕ(X) in one free variable X, written in the language of rings, such that for any prime p with gcd(a,p)=1 the set of all elements in the finite prime field Fp satisfying ϕ coincides with the range of the discrete exponential function t↦at(modp)

Sequential experiments with primes

With a specific focus on the mathematical life in small undergraduate colleges, this book presents a variety of elementary number theory insights involving sequences largely built from prime numbers and contingent number-theoretic functions. Chapters include new mathematical ideas and open problems, some of which are proved in the text. Vector valued MGPF sequences, extensions of Conway’s Subprime Fibonacci sequences, and linear complexity of bit streams derived from GPF sequences are among the topics covered in this book. This book is perfect for the pure-mathematics-minded educator in a small undergraduate college as well as graduate students and advanced undergraduate students looking for a significant high-impact learning experience in mathematics