Mihai Caragiu

Invariants for Linear Recurrences

There is no homogeneous polynomial of degree two, \(\phi (x,y) = a{{x}^{2}} + bxy + c{{y}^{2}} \) for which the classical Fibonacci sequence {F n} satisfies an identity of the form

Conway’s Subprime Function and Related Structures with a Touch of Fibonacci Flavor

\phi(F_{n},F_{n-1}) = constant

Greatest Prime Factor Sequences

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Going All Experimental: More Games and Applications

Abstract Using an exponential sum associated to the Legendre character, we introduce a finite ‘upper half-plane’ V(q), by defining a metric on the set given by the union between the quotient of Fq2 − Fq with respect to the Frobenius action, and an extra point ∞, which appears as a collapse of the field Fq. We also introduce, for every odd prime power q, the ‘length spectrum’ Σq, that is, the set of all possible distances between distinct points of V(q), which plays the role of a ‘parameter space’ for a class of associated graphs V(q; k), k ϵ Σq, for which the ‘finite parts’ Vo(q; k) are regular. Up to a normalization, the whole metric space V(q) can be seen as a small perturbation of a complete graph with 1 + (q2 − q)/2 vertices. Finally, we show how these results generalize to any higher dimension n. The corresponding metric space Vn(q) is obtained out of the set of the orbits of the Frobenius action on Fqn over Fq, by making appropriate identifications.

ON THE RANGES OF DISCRETE EXPONENTIALS

While the previous chapter was primarily devoted to recurrences involving sequences of primes, or sequences of vectors with prime components, or sequences of sets of primes in the context of algebraic structures defined on the set of primes, all of them set up in terms of the greatest prime factor function, in the present chapter we will address special recurrent sequences of integers defined in terms of two other number-theoretic functions that involve in an essential way in their definitions the prime factors of their argument. These are the Euler’s phi (or totient) function, and Conway’s subprime function.

Zero sets of polynomials: one versus two variables

The oldest known record of a mathematical object is the fossilized Lebombo bone [more than 43,000 years old according to rigorous carbon dating (d’Errico et al. 1987)], displaying 29 tally marks. Just for fun: this means that the first integer ever recorded in human history happens to be a prime number! One could indeed say that the natural numbers 1, 2, 3, … have fascinated humanity from the dawn of time, thus becoming engrained in human consciousness (Greathouse; Dehaene 1999).

Constructing irreducible polynomials with prescribed level curves over finite fields

Our involvement with “GPF sequences” began at Ohio Northern University in the fall of 2005, out of the necessity of providing one of my students (Lisa Scheckelhoff) with a topic for her senior capstone project.

The GL(n,Fp)invariance of the Potts Hamiltonian

In the first part of this chapter we will explore new types of cellular automata that emerged from our work with Ducci games based on the greatest prime factor or Conway’s subprime function.