This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be represented as $a^2+ab+b^2$ if and only if $p$ is of the form $6k+1$, with the only exception of 3, (ii) any positive integer can be represented as $a^2+ab+b^2$ if and only if its all prime factors that are not in the same form have even exponents in the standard factorization, and (iii) all the factors of an integer in the form $a^2+ab+b^2$, where $a$ and $b$ are positive and relatively prime to each other, are also of the same form. A general formula for the number of distinct representations of any positive integer in this form is conjectured. A comparison of the results with the properties of some other binary quadratic forms is given.
We develop an elementary divisor theory for the unimodular and the modular group over quadratic field extensions and quaternion algebras. In particular, we investigate which sets of elementary divisors can occur. Under an additional hypothesis we establish a correspondence of unimodular and modular double cosets.
Mordell in 1958 gave a new proof of the three squares theorem. Those techniques were generalized by Blackwell, et al., in 2016 to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries". We continue the generalization of these techniques to four additional forms.
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than the square root of p. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds the square root of p.
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…
This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.
We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum_{p^2+q^2\leq N} τ(p^2+q^2+1)$, where $p,q$ are primes.
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 purpose of this paper is to present a collection of interesting generating functions for partitions which have connections to positive definite binary quadratic forms. In establishing our results we obtain some new Bailey pairs.
We prove that for any countable set $A$ of real numbers, the set of binary indefinite quadratic forms $Q$ such that the closure of $Q(\mathbb{Z}^2)$ is disjoint from $A$ has full Hausdorff dimension.
For a congruence subgroup $Γ$, we define the notion of $Γ$-equivalence on binary quadratic forms which is the same as proper equivalence if $Γ= \mathrm{SL}_2(\mathbb Z)$. We develop a theory on $Γ$-equivalence such as the finiteness of $Γ$-reduced forms, the isomorphism between $Γ_0(N)$-form class group and the ideal class group, $N$-representation of integers, and $N$-genus of binary quadratic forms. As an application, we deal with representations of integers by binary quadratic forms under certain congruence condition on variables.
Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.
Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary f…
We state and prove an identity which represents the most general eta-products of weight 1 by binary quadratic forms. We discuss the utility of binary quadratic forms in finding a multiplicative completion for certain eta-quotients. We then derive explicit formulas for the Fourier coefficients of certain eta-quotients of weight 1 and level 47, 71, 135,648 1024, and 1872.
By an elementary observation about the computation of the difference of squares for large in- tegers, deterministic quadratic Frobenius probable prime tests are given with running times of approximately 2 selfridges.
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.
In this note we show that for a given irreducible binary quadratic form $f(x,y)$ with integer coefficients, whenever we have $f(x,y) = f(u,v)$ for integers $x,y,u,v$, there exists a rational automorphism of $f$ which sends $(x,y)$ to $(u,v)$.
In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form $F$ and binary linear form $G$, there exist infinitely many $\ell, m\in\mathbb{Z}$ such that both $F(\ell, m)$ and $G(\ell, m)$ are primes as long as there are no local obstructions.
Let $f(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with integer coefficients. For a prime $p$ not dividing the discriminant of $f$, we say $f$ is completely $p$-primitive if for any non-zero integer $N$, the diophantine equation $f(x,y)=N$ has always an integer solution $(x,y)=(m,n)$ with $(m,n,p)=1$ whenever it has an integer solution. In this article, we study various properties of completely $p$-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form $f$ to be completely $p$-primitive.
Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that this list is complete outside of "trivial" pairs. In this article, we find all pairs of such forms that represent the same primes outside of a finite set.
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by Pépin in a series of articles on unsolvable quartic diophantine equations, and for constructing families of counterexamples to the Hasse Principle for curves of genus 1 defined by equations of the form $ax^4 + by^4 = z^2$.
In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.
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
Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d) ... r_k(n+ (k-1)d).
We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.
In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also characterize, when q>13, all the definite binary forms over F_q[t] that have class number one.
Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.
We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtain by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite fa…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic f…
We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.
We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle.
InfoMod is a new software and application devoted to the modular group, PSL2(Z). It has algorithms that deals with the classical correspondences among continued fractions, geodesics on the modular surface and binary quadratic forms. In addition the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be…
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.
In this paper we study a geometric coding algorithm for indefinite binary quadratic forms Q for the congruence subgroup Γ^0(N), with respect to the usual fundamental domain FN, where N is assumed prime. The cycles Q_1, . . ., Q_n that this algorithm produces are such that the the corresponding paths γ_1, . . ., γ_n in the Riemann surface X0(N)(C) have a nice behavior around the elliptic points of order 2.
Given an integer $d \ge 2$, what is the least $r$ so that there is a set of binary quadratic forms $\{f_1,\dots,f_r\}$ for which $\{f_j^d\}$ is non-trivially linearly dependent? We show that if $r \le 4$, then $d \le 5$, and for $d \ge 4$, construct such a set with $r = \lfloor d/2\rfloor + 2$. Many explicit examples are given, along with techniques for producing others.
We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding results of Bombieri-Vinogradov type and of Barban-Davenport-Halberstam type, although with shorter ranges than in the original theorems for primes in arithmetic progressions: The results imply that, for all $a>0$, the least prime that can be represented by any given positive definite binary quadratic form of discriminant $q$ is smaller than $|q|^{7+a}$ for all for…
In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$. We also obtain a quantitative result for the number of sign changes of the sequence of the normalised Fourier coefficients $λ_{f}(n)$ of the Hecke eigenforms $f$ where $n$ is represented by any primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when the class number $h(D) = 1$ in…
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least one of $p \in S$.
We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every choice of an edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. Reduced forms in the class represented by a çark correspond to some distinguished edges on its spine. Gauss reduction is the process of moving the edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented b…
We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make extensive use of modified negative continued fractions.
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by the binary quadratic form $x^2 + 3xy + y^2$. The Euclidean algorithm is commenced with inputs from one of the families, and the first remainder less than a predetermined size produces the modular inverse or representation.
In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.
A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If there is an integer bilinear map s such that f(s(x,y))=f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f has semigroup property. We give an explicit description of all pairs (f,s) with the property stated above. We do not know any other examples of forms with semigroup property.
