# Almost-universal quadratic forms: An effective solution of a problem of Ramanujan

@article{Bochnak2009AlmostuniversalQF, title={Almost-universal quadratic forms: An effective solution of a problem of Ramanujan}, author={Jacek Bochnak and Byeong-Kweon Oh}, journal={Duke Mathematical Journal}, year={2009}, volume={147}, pages={131-156} }

The object of this paper is to prove several results giving an effective method for deciding whether a positive definite integral quaternary quadratic form is almost universal, that is, whether it represents all large positive integers. In this way we obtain an effective and definitive solution to a problem first addressed and investigated by Ramanujan 90 years ago (cf. [11]). The following set Σ = {1, 2, 3, 5, 6, 7, 10, 14}

## 14 Citations

### On primitively universal quadratic forms

- Mathematics
- 2010

In 1999, Manjul Bhargava proved the Fifteen Theorem and showed that there are exactly 204 universal positive definite integral quaternary quadratic forms. We consider primitive representations of…

### On locally primitively universal quadratic forms

- Mathematics
- 2020

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive…

### Almost universality of a sum of norms

- Mathematics
- 2013

In this paper the author considers a particular type of polynomials with integer coefficients, consisting of a perfect power and two norm forms of abelian number fields with coprime discriminants. It…

### The Kloosterman problem for binary Hermitian lattices

- Mathematics, Physics
- 2014

A Hermitian lattice over an imaginary quadratic field $$\mathbb {Q}(\sqrt{-m})$$Q(-m) is called almost universal if it represents all but finitely many positive integers. We investigate almost…

### A Characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2

- Mathematics
- 2015

### Even universal binary Hermitian lattices and an application to the Kloosterman problem over imaginary quadratic fields

- Mathematics, Computer Science
- 2008

The results to the Kloosterman problem are applied and the Bochnak-Oh type criterion on almost universality of binary Hermitian lattices over Q(√ −m) is provided.

### Representations of integral quadratic polynomials

- Mathematics
- 2012

In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral…

### On primitively 2-universal quadratic forms

- Mathematics
- 2012

The primitive representations of binary positive definite, classically integral quadratic forms over the local rings Zp are studied. For the global ring, an efficient method is obtained for…

### Classically integral quadratic forms excepting at most two values

- MathematicsProceedings of the American Mathematical Society
- 2018

Let $S \subseteq \mathbb{N}$ be finite. Is there a positive definite quadratic form that fails to represent only those elements in $S$? For $S = \emptyset$, this was solved (for classically integral…

### Almost universal sums of triangular numbers with one exception

- Mathematics
- 2022

. For an arbitrary integer x , an integer of the form T p x q “ x 2 ` x 2 is called a triangular number. For positive integers α 1 ,α 2 ,...,α k , a sum ∆ α 1 ,α 2 ,...,α k p x 1 ,x 2 ,...,x k q “ α…

## References

SHOWING 1-10 OF 12 REFERENCES

### History of the Theory of Numbers

- MathematicsNature
- 1924

THE third and concluding volume of Prof. Dickson's great work deals first with the arithmetical. theory of binary quadratic forms. A long chapter on the class-number is contributed by Mr. G. H.…

### On the Representation of Numbers in the Form ax2+by2+cz2+dt2

- Mathematics
- 1926

i . i . The object of the present paper is to treat the problem of the representation of large positive integers in the form a x Z + b y S + c z Z + d t 2 (where a, b, e, d are given positive…

### A Course in Arithmetic

- Mathematics
- 1973

Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on…

### Introduction to quadratic forms

- Mathematics
- 1963

Prerequisites ad Notation Part One: Arithmetic Theory of Fields I Valuated Fields Valuations Archimedean Valuations Non-Archimedean valuations Prolongation of a complete valuation to a finite…

### Almost regular quaternary quadratic forms

- Mathematics
- 2008

Nous etudions les formes quadratiques entieres quaternaires (c'est-a-dire a quatre variables) qui sont definies positives et presque regulieres. Nous montrons en particulier qu'une telle forme n'est…

### On the Conway-Schneeberger fifteen theorem, in Quadratic Forms and Their Applications (Dublin), pp. 27-37

- Contemporary Math., 272. Amer. Math. Soc.,
- 2000