Odd Partitions Research Articles

A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers

The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form Eij = ni + nj + (nj - ni)n or alln ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.

A Logical Proof of the Polignac's Conjecture Based on Partitions of an Even Number of a New Formulation

Polignac's Conjecture, proposed by Alphonse de Polignac in the 19th century, is a captivating hypothesis that extends the notion of twin primes to a broader context. It posits that for any even positive integer 2k, there exist infinitely many pairs of consecutive prime numbers whose difference is 2k. This conjecture is a natural generalization of the Twin Prime Conjecture, which focuses solely on pairs of primes differing by two. The conjecture has significant implications for our understanding of the distribution of prime numbers and the nature of their gaps and its exploration serves as a testament to the enduring fascination and mystery surrounding prime numbers and their properties. However, despite extensive efforts by mathematicians over the years, Polignac's Conjecture remains unproven, standing as one of the many unsolved problems in number theory. This study utilizes a set of all odd partitions generated from an even number of a new formulation, and we show that from this set of all pairs of odd numbers, there exist proper subsets containing infinitely many pairs of prime numbers whose differences is a fixed even gap. Finally, using these results and the facts that the difference of any two prime numbers is even and there exist infinitely many prime numbers, a logical proof of the Polignac's Conjecture is provided.

A short note on decay rates of odd partitions: an application of spectral asymptotics of the Neumann–Poincaré operators

A short note on decay rates of odd partitions: an application of spectral asymptotics of the Neumann–Poincaré operators

Congruence Modulo 4 for Andrews’ Even Parts Below Odd Parts Partition Function

Congruence Modulo 4 for Andrews’ Even Parts Below Odd Parts Partition Function

Even–odd partition identities of Rogers–Ramanujan type

We prove a theorem which adds a new companion to the Rogers–Ramanujan identities. This new companion counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two families of partition identities.

Bounds for coefficients of the f(q) mock theta function and applications to partition ranks

We compute effective bounds for α(n), the Fourier coefficients of Ramanujan's mock theta function f(q) utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of Hou and Jagadeesan on the convexity and maximal multiplicative properties of the even and odd partition rank counting functions.

On the partitions into distinct parts and odd parts

In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler’s recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also derived. In this context, we conjecture that for k > 0, the series has non-negative coefficients.

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with Rin hbox {SP} are common eigenfunctions of cut-and-join operators W_Delta with Delta in hbox {OP}. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a tau -function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

Generalization of two problems of George Beck

Recently, Beck posed two conjectures on the difference between the number of (respectively, distinct) parts in the odd partitions of n and the number of (respectively, distinct) parts in the distinct partitions of n. These two conjectures were first confirmed by Andrews using generating functions, and then generalized by Fu and Tang, and Yang in different ways. Motivated by Yang’s work, we present two more generalized results and prove them both analytically and combinatorially.

Combinatorial Proofs of Two Euler-Type Identities Due to Andrews

Let a(n) be the number of partitions of n, such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all odd partitions of n and the number of parts in all distinct partitions of n, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that a(n) = b(n) and Andrews, using generating functions, proved that a(n) = b(n) = c(n). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. We prove combinatorially Beck’s second conjecture, which was also proved by Andrews using generating functions. Let c1(n) be the number of partitions of n, such that there is exactly one part occurring three times, while all other parts occur only once and let b1(n) be the difference between the total number of parts in the partitions of n into distinct parts and the total number of different parts in the partitions of n into odd parts. Then, c1(n) = b1(n).

A bijection for Euler’s partition theorem in the spirit of Bressoud

For each positive integer n, we construct a bijection between the odd partitions of n and the distinct partitions of n. Our bijection extends a bijection of Bressoud between the odd-and-distinct partitions of n and the splitting partitions of n. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with one recently constructed by Chen, Gao, Ji and Li.

Spaltenstein varieties of pure dimension

We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan-type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the σ \sigma -quiver-variety setting.

Hook removal operators on the odd Young graph

In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck’s conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher’s bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.

Variation on a theme of Nathan Fine. New weighted partition identities

We utilize false theta function results of Nathan Fine to discover four new partition identities involving weights. These relations connect Göllnitz–Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition.

Scheduling with time-dependent discrepancy times

In time-dependent scheduling, various processing time functions are studied, yet absolute value functions have surprisingly been omitted from the discussion. Such a processing time function increases linearly with a job's discrepancy from its ideal midtime. The objective is to find a schedule that minimizes the makespan, introducing the discrepancy time minimization problem. This single-machine scheduling problem with time-dependent processing times is motivated by optimization of walking times on a car assembly line. Its decision version is NP hard, as we show by reduction of the even---odd partition problem. For the variant with known start time, we develop several heuristics. Further insights form lower bounds and dominance rules for a branch-and-bound search. Numerical experiments show the performance of our algorithms on problem instances of up to 60 jobs. For the variant with common ideal midtime and flexible start time, we present a polynomial-time algorithm.

Inversion of the Odd-Even Effect in Cold Fission from the Time-Dependent Pairing Equations

A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones for excitations energies of the fragments smaller than 4 MeV. In this contribution, a microscopic model is proposed for the explanation of this odd-even effect in cold fission. This explanation is based on a mixing configuration mechanism that is produced during the fission process. This configuration mixing mechanism is obtained dynamically by solving a the generalized system of time-dependent pairing equations, which include a pair-breaking effect. The time dependent equations of motion for the pair breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two fission fragment. The single particle level scheme was calculated with the Woods-Saxon superasymmetric two center shell model, providing a continuous variation of the single particle energies and of the wave functions from one nucleus up to two separated fragments. A first rule can be extracted from this model. The even-even fission products cannot be obtained at zero excitation energies because of the existence of dynamical excitations produced in the avoided- level-crossing regions when the nuclear system deforms slowly.

A Fine Rediscovery

This article explores the history of the two results in integer partitions known as Stanley’s theorem and Elder’s theorem. While history has credited Richard Stanley with the discovery of the results, we note that Nathan Fine had established these results among a host of other partition identities over a decade earlier. In tribute to Fine, analogues in the sets of odd partitions and distinct partitions are presented.

Microscopic description of the odd-even effect in cold fission

The time-dependent equations of motion for the pair-breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two-fission fragment. The single-particle level scheme was calculated with the Woods--Saxon superasymmetric two-center shell model. This model provides a continuous variation of the energies from one nucleus up to two separated fragments. The dissipated energy resorts from the time-dependent pairing equations. A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones. This odd-even effect for cold fission was explained microscopically.

On Compatible Normal Odd Partitions in Cubic Graphs

On Compatible Normal Odd Partitions in Cubic Graphs