Almost universal ternary sums of polygonal numbers

Abstract

For a natural number m, generalized m-gonal numbers are those numbers of the formwith x ∈ Z. In this paper we establish conditions on m for which the ternary sum p m (x) + p m (y) + p m (z) is almost universal.

Near-miss Identities and Spinor Genus Classification of Ternary Quadratic Forms with Congruence Conditions

“…A further generalization of equation (1.2) has also been conjectured in [4], which may be stated as 1…”

“…the number of spinor genera), we can establish the decomposition into spinor genera. This general strategy is used in [9] as well as in [4].…”

“…Representations with congruence conditions are useful, for instance, in the study of quadratic polynomials, which are generalizations of quadratic forms with a non-zero linear contribution of x. By completing squares, many quadratic polynomials may be transformed into quadratic forms under congruence conditions (see for instance [2] and [4]).…”

The Bruinier–Funke Pairing and the Orthogonal Complement of Unary Theta Functions

“…In the case of triangular numbers (that is to say, m = 3), the first author and Sun [10] obtained a nearclassification which was later fully resolved by Chan-Oh [5]; further classification results about sums of triangular numbers and squares were completed by Chan-Haensch [4]. More recently, the case a = b = c = 1 with arbitrary m was considered by Haensch and the first author [8]. In [8], a number of almost universality results are obtained by taking advantage of the fact that the structure of modular forms may be used to determine that certain congruence classes are not in the support of the coefficients of all of the unary theta functions in the same space, and hence directly obtaining the orthogonality needed for Theorem 1.1.…”

Almost universal ternary sums of pentagonal numbers

“…For convenience, a quadratic polynomial is called almost universal if it can represent all but finitely many positive integers over Z. For the recent work on almost universal quadratic polynomials, readers may refer to [2,3,8,9,10,12,16,22,23,24]. Throughout this paper, we assume that the following condition ( * ) is satisfied:…”