Related papers: Gauss-Euler Primality Test
To assess whether there is some signal in a big database, aggregate tests for the global null hypothesis of no effect are routinely applied in practice before more specialized analysis is carried out. Although a plethora of aggregate tests…
We use Experimental Mathematics and Symbolic Computation (with Maple), to search for lots and lots of Perrin- and Lucas- style primality tests, and try to sort the wheat from the chaff. More impressively, we find quite a few such primality…
We show that every $N \geq 2$ can be written as the sum of positive integers $a$ and $b$ where $\Omega(ab) \leq 40$. The result is obtained through the direct application of an explicit lower bound Selberg sieve along with some computation…
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second…
We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity…
Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent…
Non-degenerate U-empirical Kolmogorov-Smirnov tests are studied and their large deviation asymptotics under the null-hypothesis is described. Several examples of such statistics used for testing goodness-of-fit and symmetry are considered.…
We study those $(2,m,n)$-groups which are almost simple and for which the absolute value of the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely…
We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress…
The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and…
In this paper we propose a conjecture about integer solutions to any equations, based on Primal algebra specifically this conjecture is a corollary of the Acu\~na Theorem in that article. Also some problems are proposed which, if the…
We present an iterative method to solve the multipartite quantum state estimation problem. We demonstrate convergence for any informationally complete set of generalized quantum measurements in every finite dimension. Our method exhibits…
The timing of radio pulsars in binary systems provides a superb testing ground of general relativity. Here we propose a Bayesian approach to carry out these tests, and a relevant efficient numerical implementation, that has several…
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…
We propose a novel kernel-based nonparametric two-sample test, employing the combined use of kernel mean and kernel covariance embedding. Our test builds on recent results showing how such combined embeddings map distinct probability…
We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem} (are all but finitely many terms of a…
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 study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0, 2, 4, 6, 8\}$. Computational checks up to $n = 10^{15}$ reveal the known cases $n = 1, 2, 3, 6, 11$ and no…
The integer complexity $f(n)$ of a positive integer $n$ is defined as the minimum number of 1's needed to represent $n$, using additions, multiplications and parentheses. We present two simple and faster algorithms for computing the integer…