Related papers: Strong Pseudo Primes to Base 2
We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have manifold applications…
We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra…
Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…
We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…
For a restricted Lie superalgebra g over an algebraically closed field of characteristic p > 2, we generalize the deformation method of Premet and Skryabin to obtain results on the p-power and 2-power divisibility of dimensions of…
The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem…
This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power,…
In this paper, we establish a Bombieri-Vinogradov type result for prime numbers of the form $p=x^2+y^2+1$. The proof is based on the enveloping sieve.
This paper reformulates and streamlines the core tools of robust stability and performance for LTI systems using now-standard methods in convex optimization. In particular, robustness analysis can be formulated directly as a primal convex…
We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.
Following Stolarsky, we say that a natural number n is flimsy in base b if some positive multiple of n has smaller digit sum in base b than n does; otherwise it is sturdy. We develop algorithmic methods for the study of sturdy and flimsy…
In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…
In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…
In this paper, we study the perturbation analysis of a class of composite optimization problems, which is a very convenient and unified framework for developing both theoretical and algorithmic issues of constrained optimization problems.…
An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The…
Weak superimposed codes are combinatorial structures related closely to generalized cover-free families, superimposed codes, and disjunct matrices in that they are only required to satisfy similar but less stringent conditions. This class…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
We establish a necessary condition for pseudoprimality and a sufficient condition for primality of Fermat numbers, based on a congruence involving the exponent $(F_n-1)/4$. Moreover, in connection with P\'epin's primality test, we obtain a…
The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…