# 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}

## 12 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…

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.

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…

Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

- Mathematics
- 2022

Recently, Rattan and the ﬁrst author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible…

Geometry of Numbers with Applications to Number Theory

- Mathematics
- 2012

1. Lattices in Euclidean Space 3 1.1. Discrete vector groups 3 1.2. Hermite and Smith Normal Forms 5 1.3. Fundamental regions, covolumes and sublattices 6 1.4. The Classification of Vector Groups 12…

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