Composite Integer Research Articles

Coprime partitions and Jordan totient functions

We show that while the number of coprime compositions of a positive integer n into k parts can be expressed as a Q-linear combination of the Jordan totient functions, this is never possible for the coprime partitions of n into k parts. We also show that the number pk′(n) of coprime partitions of n into k parts can be expressed as a C-linear combination of the Jordan totient functions, for n sufficiently large, if and only if k∈{2,3} and in a unique way. Finally we introduce some generalizations of the Jordan totient functions and we show that pk′(n) can be always expressed as a C-linear combination of them.

More Connections on Valuated Binary Tree and Their Applications in Factoring Odd Integers

This paper continues investigating connections on a valuated binary tree. By defining three types of new connections, the paper derives several new properties for the new connections, and proves that odd integers matching to the new cases can be easily and rapidly factorized. Proofs are presented for the new properties and conclusions with detail mathematical reasoning and numerical experiments are made with Maple software to demonstrate the fast factorization by factoring big odd composite integers that are of the length from 101 to 105 decimal digits. Source codes of Maple programs are also list for readers to test the experiments.

Locally restricted compositions over a finite group

One may generalize integer compositions by replacing the positive integers with a different additive semigroup, giving the broader concept of a “composition over a semigroup”. Here we focus on semigroups which are finite groups and achieve asymptotic enumeration of compositions over a finite group which satisfy a local restriction. These compositions are associated to walks on a voltage graph whose structure is exploited to simplify asymptotic expressions. Specifically, we show that under mild conditions the number of locally restricted compositions of a group element is asymptotically independent of the particular group element. We apply this result to subword pattern avoidance and other examples such as generalized Carlitz compositions.

Prolific Permutations


 The concept of prolificity was previously introduced by the authors in the context of compositions of integers. We give a general interpretation of prolificity that applies across a range of relational structures defined in terms of counting embeddings. We then proceed to classify prolificity in permutation classes with bases consisting of permutations of length 2, or 3; completely classifying all such classes except ${\rm Av}(321)$. We then show a number of interesting properties that arise when studying prolificity in ${\rm Av}(321)$, concluding by showing that the class of permutations that are not prolific for any increasing permutation in ${\rm Av}(321)$ form a polynomial subclass.

Duality of Graded Graphs Through Operads

Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use operads, that algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric operads. This pair of graphs is dual for a new notion of duality called $\phi$-diagonal duality, similar to the ones introduced by Fomin. We also provide a general way to build pairs of graded graphs from operads, wherein underlying posets are analogous to the Young lattice. Some examples of operads leading to new pairs of graded graphs involving integer compositions, Motzkin paths, and $m$-trees are considered.

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.

Absolute irreducibility of the binomial polynomials

In this paper we investigate the factorization behaviour of the binomial polynomials (xn)=x(x−1)⋯(x−n+1)n! and their powers in the ring of integer-valued polynomials Int(Z). While it is well-known that the binomial polynomials are irreducible elements in Int(Z), the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int(Z), that is, (xn)m factors uniquely into irreducible elements in Int(Z) for all m∈N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n>10 and n, n−1, …, n−(k−1) are composite integers, then there exists a prime number p>2k that divides one of these integers.

On sum of prime factors of composite positive integers

Let $${\mathfrak {B}}(x)$$ be the number of composite positive integers up to x whose sum of distinct prime factors is a prime number. Luca and Moodley proved that there exist two positive constants $$a_1$$ and $$a_2$$ such that $$\begin{aligned} a_1x/\log ^3x\le {\mathfrak {B}}(x)\le a_2x/\log x. \end{aligned}$$ Assuming a uniform version of the Bateman–Horn conjecture, they gave a conditional proof of a lower bound of the same order of magnitude as the upper bound. In this paper, we offer an unconditional proof of the this result, i.e., $$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$

How to detect whether Shor’s algorithm succeeds against large integers without a quantum computer

Shor’s algorithm is a well-known probabilistic method for factoring large composite integers in polynomial-time on a quantum computer. The method computes the order r of a random element x in the group Z∗N and uses that information for splitting N with an application of the greatest common divisor algorithm. However, being probabilistic, the success of Shor’s algorithm relies on some special properties of N. If r is even and xr/2 £ -1 mod N, then gcd(xr/2 - 1, N) reveals a nontrivial factor of N and the method succeeds. But even assuming that r is even and being given the complete prime factorization of N it is not obvious whether xr/2 £ -1 mod N and, therefore, it is not easy to assert whether Shor’s algorithm would split N without running it and looking at its answer. We present a strategy for detecting whether the splitting occurs without any need for running the quantum order-finding algorithm, but we must be given the prime factorization of N. This has allowed us to produce the first direct evidence of the probability of success of Shor’s method. The composites chosen were the product of two randomly-generated probable primes of similar sizes that pass the Miller-Rabin test.

Moderate parts in regenerative compositions: The case of regular variation

A regenerative random composition of integer n is constructed by allocating n standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator S. Assuming that the Lévy measure of S is infinite and regularly varying at zero of index −α, α∈(0,1), we find an explicit threshold r=r(n), such that the number Kn,r(n) of blocks of size r(n) converges in distribution without any normalization to a mixed Poisson distribution. The sequence (r(n)) turns out to be regularly varying with index α/(α+1) and the mixing distribution is that of the exponential functional of S. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of Kn,w(n) in cases when w(n) diverges but grows slower than r(n). Our findings complement previously known strong laws of large numbers for Kn,r in case of a fixed r∈N. As a key tool we employ new Abelian theorems for Laplace–Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

The Prime state and its quantum relatives

The Prime state of n qubits, |Pn⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

Integer factoring and compositeness witnesses

AbstractWe describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp $\begin{array}{} \displaystyle \left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right) \end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored.

Shift-plethystic trees and Rogers–Ramanujan identities

By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers–Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers–Ramanujan continued fraction. By specializing the non-commutative series to q-series, we obtain new combinatorial interpretations of the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative series and use this to obtain interesting enumerative identities involving compositions and partitions related to Rogers–Ramanujan identities.

Faster initial splitting for small characteristic composite extension degree fields

Let p be a small prime and n=n1n2>1 be a composite integer. For the function field sieve algorithm applied to Fpn, Guillevic (2019) had proposed an algorithm for initial splitting of the target in the individual logarithm phase. This algorithm generates polynomials and tests them for B-smoothness for some appropriate value of B. The amortised cost of generating each polynomial is O(n22) multiplications over Fpn1. In this work, we propose a new algorithm for performing the initial splitting which also generates and tests polynomials for B-smoothness. The advantage over Guillevic splitting is that in the new algorithm, the cost of generating a polynomial is O(nlogp⁡(1/π)) multiplications in Fp, where π is the relevant smoothness probability.

On the distribution of quadratic residues and non-residues modulo composite integers and applications to cryptography

We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form a+X={(a+x)modn∣x∈X}, where n is a prime or the product of two primes and X is a subset of integers with given Jacobi symbols modulo prime factors of n. We then present applications of these formulas to Cocks’ identity-based encryption scheme and statistical indistinguishability.

The Hopf monoid of orbit polytopes

Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.

Fast Approach to Factorize Odd Integers with Special Divisors

The paper proves that an odd composite integer N can be factorized in O((log2N)4) bit operations if N = pq, the divisor q is of the form 2αu +1 or 2αu-1 with u being an odd integer and α being a positive integer and the other divisor p satisfies 1 < p ≤ 2α+1 or 2α +1 < p ≤ 2α+1-1. Theorems and corollaries are proved with detail mathematical reasoning. Algorithm to factorize the odd composite integers is designed and tested in Maple. The results in the paper demonstrate that fast factorization of odd integers is possible with the help of valuated binary tree.

Counting Water Cells in Pattern Restricted Compositions

In this paper, we consider statistics on compositions of a positive integer represented geometrically as bargraphs that avoid certain classes of consecutive patterns. A unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within its subtended area is called a water cell since it is a place where a liquid would collect if poured along the top part of the bargraph from above. The total number of water cells in the bargraph representation of a k-ary word then gives what is referred to as the capacity of w. Here, we determine the distribution of the capacity statistic on certain pattern-restricted compositions, regarded as k-ary words. Several general classes of patterns are considered, including and where a is arbitrary. As a consequence of our results, we obtain all of the distinct distributions for the capacity statistic on avoidance classes of compositions corresponding to 3-letter patterns having at most two distinct letters. Finally, in the case of some further enumerative results are given when a=2, including algebraic and bijective proofs for the total capacity of all Carlitz partitions of a given size having a fixed number of blocks.

Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms

All geometrically distinct nets (edge-unfoldings) are explicitly constructed for the infinite family of polyhedra known as uniform antiprisms. The Fibonacci numbers are shown to count the number of nets of the n-antiprism that have point symmetry. (The n-antiprism refers to the antiprism whose surface consists of two regular n-gons separated by a band of 2n equilateral triangles.) Smaller evenly indexed Fibonacci numbers count certain subfamilies of these nets. Well-known formulas for sums of alternately indexed Fibonacci numbers motivate and become accessories to our constructions and proofs. For counting and constructing the symmetric nets, we first use a known result connecting compositions of integers and Fibonacci numbers. In an alternative proof, we illustrate a striking self-similarity among the counts of certain natural subfamilies of the symmetric nets, which is used to generate an efficient recursive construction of all the symmetric nets of the n-antiprism from those of the -antiprism. (This second proof is found in the online supplement to this article.) Finally, each nonsymmetric net of the n-antiprism is constructed by combining a pair of distinct symmetric nets of the n-antiprism.