Related papers: The non-existence of cubic Legendre multiplier seq…
In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship…
This research announcement describes in very rough terms methods and a computer language under development, which can be used to prove the nonexistence of binary linear codes. Over a hundred new results have been obtained by the author. For…
We prove that every multiplier sequence for the Legendre basis which can be interpolated by a polynomial has the form $\{h(k^2+k)\}_{k=0}^{\infty}$, where $h\in\mathbb{R}[x]$. We also prove that a non-trivial collection of polynomials of a…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the…
In this paper we prove a conjecture that $D(4)$-quintuple does not exist using both classical and new methods. Also, we give a new version of the Rickert's theorem that can be applied on some $D(4)$-quadruples.
The Erd\H{o}s-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
We give a new proof of the fact that Barker polynomials of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence…
This note concerns the non-existence of three consecutive powerful numbers. We use Pell equations, elliptic curves, and second-order recurrences to show that there are no such triplets with the middle term a perfect cube and each of the…
In this paper we give the first method for constructing n-multimagic squares (and hypercubes) for any n. We give an explicit formula in the case of squares and an effective existence proof in the higher dimensional case. Finally we prove…
In this paper we present a complete characterization of geometric and linear multiplier sequences for generalized Laguerre bases. In addition, we give a partial characterization of the generic multiplier sequence for such bases, and pose…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the…
In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine…
Recently the problem of constructing a perfect Euler cuboid was related with three conjectures asserting the irreducibility of some certain three polynomials depending on integer parameters. In this paper a partial result toward proving the…
We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
In this note, we give a new nonexistence result of ternary extremal self-dual codes.
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,…
We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods).
Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related…