A New Proof of the Hansen—Mullen Irreducibility Conjecture

A New Proof of the Collatz Conjecture

This paper is a revised version of a previous paper by the author that, after publication, was found to contain an error. This paper supersedes and replaces the previous paper. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N0 to the Collatz Conjecture is hypothesized ; the existence of N0 implies that N...

A new proof of Friedman's conjecture

We give a new proof of Friedman’s conjecture that every uncountable ∆1 set of reals has a member of each hyperdegree greater than or equal to the hyperjump.

A New Short Proof of Kneser's Conjecture

A new proof of Vázsonyi's conjecture

We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n − 2. The proof avoids the ball polytopes used in the original proofs by Grünbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane. Let S be a set of n points of diameter D in Rd. Define the diamet...

A short proof of the maximum conjecture in CR dimension one

In this paper and by means of the extant results in the Tanaka theory, we present a very short proof in the specific case of CR dimension one for Beloshapka's maximum conjecture. Accordingly, we prove that each totally nondegenerate model of CR dimension one and length >= 3 has rigidity. As a result, we observe that the group of CR automorphisms associated with each of such models contains onl...