Composition of binary quadratic forms

Composition of binary quadratic forms over an arbitrary commutative base ring is shown to be closely related to homomorphisms (and in particular isomorphisms) of the corresponding even Clifford algebras.     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

Prime sieves using binary quadratic forms

We introduce an algorithm that computes the prime numbers up to using additions and bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms.

     

Diophantine inequality involving binary forms

Let r =2^d-1 ＋ 1. We investigate the diophantine inequality
｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,
where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.     

Spinor genera of binary quadratic forms

Spinor genera are defined for binary quadratic forms with integer coefficients in such a way that the theory fits in with the Gaussian theory of genera. It is shown that spinor generic characters exist which distinguish the various spinor genera in the principal genus, and how they can be determined. It is known that each ambiguous class contains exactly two forms of the type [a, 0, c] or [a, a, c], each with its associate [c, 0, a], [4c ? a, 4c ? a, c]. Since the principal class contains such a form with a = 1, it is an interesting question whether one can predict the second form (not counting associates). This question includes that of Dirichlet about the representability of ?1 by the principal class. Methods are given for evaluating the spinor-generic characters of ambiguous forms in the principal genus for variable discriminants d, and are carried through in the eleven cases where d is fundamental, there are two or four genera, and two spinor genera in the principal genus. The problem of determining the “second form” is thus completely solved except when there is more than one ambiguous class in the principal spinor genus.     

Primes represented by binary quadratic forms

In 1882 Weber showed that any primitive binary quadratic form with integral coefficients represents infinitely many primes in any arithmetic progression consistent with the generic characters of the form. In this paper it is shown that for any two primitive integral binary quadratic forms with unequal but fundamental discriminants, there is an infinite set of prime numbers p in any arithmetic progression consistent with the generic characters of the forms such that both forms represent p.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Goldbach’s ternary problem involving prime numbers expressible by given quadratic forms

In this paper, we solve Goldbach’s ternary problem involving primes expressible by given primitive positive definite binary quadratic forms whose discriminants coincide with the discriminants of imaginary quadratic fields in which quadratic forms split into linear multipliers.     

Class numbers of indefinite binary quadratic forms

We determine the asymptotic average sizes of the class numbers of indefinite binary quadratic forms when ordered by the sizes of their corresponding fundamental units. The proofs make use of the Selberg trace formula.     

On character sums of binary quadratic forms

We establish character sum bounds of the form

     

The minima of indefinite binary quadratic forms

The study of the minima of indefinite binary quadratic forms has a long history and the classical results concerning the determination of such minima are stated in terms of the continued fraction expansion of the roots. These results are recast in geometric terms. Using this, and well-known geometric properties of the modular group, some necessary and sufficient conditions for a certain class of quadratic forms to have positive unattained minima are obtained.     

Arithmetic-analytic properties of binary positive-definite quadratic forms

One considers the arithmetic and analytic properties of positive-definite binary quadratic forms of discriminant –Dn². The arithmetic structure of the set of these forms is described by forms of discriminant –D and by the Hecke operators T(n). One gives some arithmetic and analytic consequences.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 5–20, 1985.