Related papers: Algorithmic Arithmetic Fewnomial Theory I: One Var…

The paper has been withdrawn by the author due to a gap in Proof of Theorem 1.1.

Let $F$ be a number field and $\mathcal{O}_F$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the unit equation $\lambda + \mu = 1$, $\lambda, \mu \in…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

We withdraw this paper due to insufficient arguments in the derivation of Theorem 1. See quant-ph/0005062 for the new paper

The famous G\"odel incompleteness theorem says that for every sufficiently rich formal theory (containing formal arithmetic in some natural sense) there exist true unprovable statements. Such statements would be natural candidates for being…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

This paper has been withdrawn because the result turns out to be trivial.

We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main…

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…

We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent…

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…

We calculate the 1/N corrections to the probability distributions of quadratic discrepancies for sets of N random points. This is achieved by the introduction of fermionic variables. We give the diagrammatic expansion up to and including…

This paper has been withdrawn by the author due to a crucial error.

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight…

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields F and the decidability of the set of irreducible polynomials over F. We adapt these theorems to the setting of differential fields K, with constrained…

This paper has been withdrawn by the authors. There was an erroneous estimate of the degree of a transformed polynomial, making the method appear more effective than it really is. We thank an anonymous referee for pointing out this error.

The paper has benn withdrawn because the computation of the external virial contains an error which invalidate the main result.

The paper has been withdrawn by the author due to a crucial error.

This paper has been withdrawn by the author due to an error in section 7. There is a new version: arXiv:1011.3352.