A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k \geq 2$, there exists a positive integer $W(k)$ such that every sufficiently large multiple of $E_k := \gcd(2^k - 1, k)$ is the sum of at most $W(k)$ binary $k$'th powers. (The hypothesis of being a multiple of $E_k$ cannot be omitted, since we show that the $\gcd$ of the binary $k$'th powers is $E_k$.) Also, we explain how our results can be extended to arbitrary integer bases $b > 2$.
Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Waring problem in numbers of class $\mathbb{N}_0$ is solved.
The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We completely classify binary forms of Waring rank 3.
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring's problem to an approximation formula for the fourier transform of the solution set of lattice points on hypersurfaces arising in Waring's problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation form…
We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function $H(k)$ in the Waring-Goldbach problem. We obtain new results for all exponents $k \ge 7$, and in particular establish that for large $k$ one has \[H(k)\le (4k-2)\log k-(2\log 2-1)k-3.\]
Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s.
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^4+p_6^b \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(b)}$ and the other variables primes, where $r(b)$ is defined in the Theorem. This result constitutes an improvement upon that of Lü and Mu.
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan for ternary admissible exponent. Moreover, we use the refined admissible exponent to prove that, for $3\leqslant k\leqslant 14$ and for every sufficiently large even integer $n$, the following equation \begin{equation*} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(k)}$ and the other variables primes, where $r(k)$ is defined in Theorem. This result constitutes a deepening…
We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ can be written as a sum of three squares. Hensel's Theorem for $p$-adic integers plays a key role. Final…
Denote by P(K, k) the members of the field K which are sums of kth powers of field elements, by P+(K, k) the set of members of K which are sums of kth powers of totally positive elements of K. We are interested in deciding whether or not there exist integers w(K, k) and g(K, k) such that: a in P(K, k) implies that a is the sum of at most w(K, k) kth powers; a in P+(K, k) implies that a is the sum of at most g(K, k) totally positive kth powers. We will show that if w(K, 2) is finite and provided that the kth powers are dense (in a sense described explicitly in theorem 3) in K, then w(K, k) is a…
We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.
In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for t…
With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.
Explicit counterexamples to Theorem 1 of R. Turyn's and J. Storer's often-cited paper "On Binary Sequences" [Proc. Amer. Math. Soc., vol. 12, no. 3, pp. 394-399, 1961] are given. Theorem 2 of their paper is well known; it states that Barker sequences of odd length n>13 do not exist. Since the proof of Theorem 2 relies on the incorrect Theorem 1, the proof of the often-cited Theorem 2 is therefore not correct. It is not at all clear how Theorem 1 and its proof must be modified in order to get a correct proof of the well-known Theorem 2.
In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.
Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.
Kaplansky [2003] proved a theorem on the simultaneous representation of a prime $p$ by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink's list. We also show how such theorems are related to results of Muskat on binary quadratic forms
We previously proposed the first nontrivial examples of a code having support $t$-designs for all weights obtained from the Assmus-Mattson theorem and having support $t'$-designs for some weights with some $t'>t$. This suggests the possibility of generalizing the Assmus-Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus-Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code $\mathcal{G}_{24}$.
Let $\mathcal{K}(a)$ denote the Kloosterman sum on the finite field of order $2^n$. We give a simple characterization of $\mathcal{K}(a)$ modulo 16, in terms of the trace of $a$ and one other function. We also give a characterization of $\mathcal{K}(a)$ modulo 64 in terms of a different function, which we call the lifted trace. Our proofs use Fourier analysis, Stickelberger's theorem and the Gross-Koblitz formula.
We show how to prove theorems in additive number theory using a decision procedure based on finite automata. Among other things, we obtain the following analogue of Lagrange's theorem: every natural number > 686 is the sum of at most 4 natural numbers whose canonical base-2 representation is a binary square, that is, a string of the form xx for some block of bits x. Here the number 4 is optimal. While we cannot embed this theorem itself in a decidable theory, we show that stronger lemmas that imply that the theorem can be embedded in decidable theories, and show how automated methods can be us…
In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.
In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier's collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof à la Racah, namely, only using the Chu-Vandermonde Theorem, for Dixon's Summation Theorem. We also provide explici…
In the present paper, we give Assmus--Mattson type theorems for codes and lattices. We show that a binary doubly even self-dual code of length 24m with minimum weight 4m provides a combinatorial 1-design and an even unimodular lattice of rank 24m with minimum norm 2m provides a spherical 3-design. We remark that some of such codes and lattices give t-designs for higher t. As a corollary, we give some restrictions on the weight enumerators of binary doubly even self-dual codes of length 24m with minimum weight 4m. Ternary and quaternary analogues are also given.
Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.
We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.
The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic f…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
We provide a necessary and sufficient condition for $n!$ to be a sum of three squares. The condition is based on the binary representation of $n$ and can be expressed by the operation of an automaton.
We give an improved estimate for the density of $k$-free values of integral binary forms with no fixed $k$-th power divisor. Further, we give the corresponding improvement to a theorem of Stewart and Top on the number of power-free values in an interval that may be assumed by a binary form. The approach we use involves a generalization of the global determinant method of Salberger.
We give two simple algorithms for the evaluation of difference between the numbers of multiples of 3 with even and odd binary digit sums in interval [0,x), and give an elementary proof of Coquet's sharp estimates for it.
The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in some depth. In this paper, we show previously unknown connections between BSD representations and the Stern polynomial. We derive a weight-distribution theorem for $i$-bit BSD representations of an integer $n$ in terms of the coefficients and degrees of the terms of the Stern polynomial of $2^i-n$. We then show new recursions on Stern polynomials, and from the…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the pr…
It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjug…
