Integers

We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .

A set of positive integers $A \subset \mathbb{Z}_{>0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{>0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{>0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$

A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function π_2(x), the number of semiprimes less than or equal to x, is studied. Using a combinatorial argument, asymptotic series of π_2(x) is determined, with all the terms explicitly given. An algorithm for the calculation of the constants involved in the asymptotic series is presented and the constants are computed to 20 significant digits. The errors of the partial sums of the asymptotic series are investigated. A generalization of this approach to products of k primes, for k≥ 3, is also proposed.

This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky's cluster algebras of type A. One can derive from the matchings model an enumerative meaning for the Markoff numbers, and prove that the associated Laurent polynomials have positive coefficients as was conjectured (m uch more generally) by Fomin and Zelevinsky. Most of this research was conducted under the auspices of REACH (Research Experiences in Algebraic Combinatorics at Harvard).

For a fixed $b\in \mathbb{N}=\{1,2,3,\dots\}$, Goins et al. \cite{Harris} defined the concept of $b$-visibility for a lattice point $(r,s)$ in $L=\mathbb{N}\times \mathbb{N}$ which states that $(r,s)$ is $b$-visible from the origin if it lies on the graph of $f(x)=ax^b$, for some positive $a\in \mathbb{Q}$, and no other lattice point in $L$ lies on this graph between $(0,0)$ and $(r,s)$. Furthermore, to study the density of $b$-visible points in $L$ Goins et al. defined a generalization of greatest common divisor, denoted by $\gcd_b$, and proved that the proportion of $b$-visible lattice points in $L$ is given by $1/\zeta(b+1)$, where $\zeta(s)$ is the Riemann zeta function. In this paper we study the mean values of arithmetic functions $\Lambda:L\to \mathbb{ C}$ defined using $\gcd_b$ and recover the main result of \cite{Harris} as a consequence of the more general results of this paper. We also investigate a generalization of a result in \cite{Harris} that asserts that there are arbitrarily large rectangular arrangements of $b$-visible points in the lattice $L$ for a fixed $b$, more specifically, we give necessary and sufficient conditions for an arbitrary rectangular arrangement containing $b$-visible and $b$-invisible points to be realizable in the lattice $L$. Our result is inspired by the work of Herzog and Stewart \cite{Herzog} who proved this in the case $b=1$.

The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest positive integer $i$ such that the $i$th iterate of Euler's totient function at $n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all numbers at a height. We state a formula for the height function due to Shapiro and use it to list steps to generate numbers at any height. This turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. Hegarty used a nontrivial algorithm to find that $8$ is the smallest cardinality of a sum-dominant set. Since then, Nathanson has asked for a human-understandable proof of the result. However, due to the complexity of the interactions among numbers, it is still questionable whether such a proof can be written down in full without computers' help. In this paper, we present a computer-free proof that a sum-dominant set must have at least $7$ elements. We also answer the question raised by the author of the current paper et al about the smallest sum-dominant set of primes, in terms of its largest element. Using computers, we find that the smallest sum-dominant set of primes has $73$ as its maximum, smaller than the value found before.

We consider differences of one- and two-variable finite products and provide combinatorial proofs of the nonnegativity of certain coefficients. Since the products may be interpreted as generating functions for certain integer partitions, this amounts to showing a partition inequality. This extends results due to Berkovich-Garvan and McLaughlin. We then apply the first inequality and Andrews' Anti-telescoping Method to give a solution to an 'Ehrenpreis Problem' for recently conjectured sum-product identities of Kanade- Russell. That is, we provide some evidence for Kanade-Russell's conjectures by showing nonnegativity of coefficients in differences of product-sides as Andrews-Baxter and Kadell did for the product sides of the Rogers-Ramanujan identities.

We give two explicit formulas for the Bernoulli numbers, one in terms of the Stirling numbers of the second kind, and the other in terms of the Eulerian numbers. To the best of our knowledge, these formulas are new. We also derive two additional formulas that are likely already known.

In the base phi expansion any natural number is written uniquely as a sum of powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we show that the sum of digits function modulo 2 of these expansions is a morphic sequence. In particular we prove that — like for the Thue-Morse sequence — the frequency of 0’s and 1’s in this sequence is equal to 1/2.

The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[\rho]$ where $\rho$ is the primitive third root of unity $e^{2\pi i/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[\rho]/(\theta)$ for any given Eisenstein integer $\theta$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[\rho]/(\gamma^n)\right)^\times$ are cyclic where $\gamma$ is prime in $\mathbb{Z}[\rho]$ and $n \in \mathbb{N}$, thereby generalizing well-known results of similar applications in the integers and some lesser known results in the Gaussian integers. As another application, we prove that the celebrated Euler-Fermat theorem holds for the Eisenstein integers.

It is the main purpose of this paper to reconsider some bivariate convolution identities discovered by Pan and Sun (and later by Zagier) for Bernoulli polynomials that involve two di↵erent types of sums (precisely, an ordinary sum and a binomial sum) and rephrase them into more concise and convincing forms of identities based on operation methods for the generating function. Furthermore, we extend one of them to a third-order trivariate convolution identity.