Related papers: Extending Babbage's (Non-)Primality Tests

Consider a problem of predicting a response variable using a set of covariates in a linear regression model. If it is \emph{a priori} known or suspected that a subset of the covariates do not significantly contribute to the overall fit of…

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

An imprecise Bayesian nonparametric approach to system reliability with multiple types of components is developed. This allows modelling partial or imperfect prior knowledge on component failure distributions in a flexible way through…

It is often claimed that Bayesian methods, in particular Bayes factor methods for hypothesis testing, can deal with optional stopping. We first give an overview, using elementary probability theory, of three different mathematical meanings…

In 1977 Pohst conjectured a certain inequality for $n$ variables and give a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms…

Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple…

Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.

We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows us to deduce complete theories of given models. The first instance of such rules was the $\omega$-rule introduced by Hilbert,…

The paper by Mayo claims to provide a new clarification and critique of Birnbaum's argument for showing that sufficiency and conditionality principles imply the likelihood principle. However, much of the arguments go back to arguments made…

Testing of various classes of life distributions has been addressed in the literature for more than 45 years. In this paper, we consider the problem of testing exponentiality (which essentially implies no ageing) against positive ageing…

We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional…

We show that many principles of first-order arithmetic, previously only known to lie strictly between $\Sigma_1$-induction and $\Sigma_2$-induction, are equivalent to the well-foundedness of $\omega^\omega$. Among these principles are the…

We present a variant of the Agrawal-Biswas algorithm, a Monte Carlo algorithm which tests the primality of an integer $N$ by checking whether or not $(x+a)^N$ and $x^N + a$ are equivalent in a residue ring of $\mathbb{Z}/N\mathbb{Z}[x]$.…

We classify essential algebras whose irredundant non-refinable covers consist of primal algebras. The proof is obtained by constructing one to one correspondence between such algebras and partial orders on finite sets. Further, we prove…

The likelihood principle makes strong claims about the nature of statistical evidence but is controversial. Its claims are undermined by the existence of several examples that are assumed to show that it allows, with unity probability,…

We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of…

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except…

We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by Dani, Laurent, and…