In this paper, we provide a generalization of Proth’s theorem for integers the form Kp^n+1. particular, primality test that requires modular exponentiation (with proper base a) similar to Fermat’s without computation any GCD’s. We also two tests increase chances proving Kp^n+1 primes. As corollaries, three families N whose can be certified only by a^{N-1} \equiv 1 \pmod (Fermat’s test). One the...

The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.

ACCS is a conservative extension of CCS, enriched with an operator of strong prefixing, enabling the modeling of atomic sequences and multi-party synchronization (realized as an atomic sequence of binary synchronizations); the classic dining philosophers problem is used to illustrate the approach. A step semantics for ACCS is also presented directly as a labeled transition system. A safe Petri ...

The next-generation Airborne Collision Avoidance System (ACAS X) is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration (FAA). In this paper we determine the geometric configurations under which the advice given by ACAS X is safe under a precise set of a...

Most theorem provers are either hard to trust because they are large programs or hard to use due to a lack of automation. They are thus used by two di erent communities depending on the property one needs most: the ability to prove quickly a large amount of theorems or a high level of safety. My thesis studies a communication between these di erent kinds of tools, by asking for proof witnesses ...

Tarski-Givant’s map calculus is briefly reviewed, and a plan of research is outlined aimed at investigating applications of this ground equational formalism in the theorem-proving field. The main goal is to create synergy between first-order predicate calculus and the map calculus. Techniques for translating isolated sentences, as well as entire theories, from first-order logic into map calculu...

The next-generation Airborne Collision Avoidance System (ACAS X) is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration (FAA). In this paper we determine the geometric configurations under which the advice given by ACAS X is safe under a precise set of a...

Tools for a specification language can be implemented directly (by building a special purpose theorem prover) or by a conservative embedding into a typed meta-logic, which allows their safe and logically consistent implementation and the reuse of existing theorem prover engines. For being useful, the conservative extension approach must provide derivations for several thousand “folklore” theore...