A subset A subset of Z is called s-almost square universal if every sufficiently large positive integer can be written as a sum of at most s squares of integers from A. In this paper, we study the minimal number ASU(A(d,m)) with this property, where A(d,m) denotes the residue class of d modulo m, with m is an element of N and d is an element of Z. We further prove that A(d,m )is s-square universal for some s is an element of N if and only if d equivalent to +/- 1 (mod m), and determine the minimal such number SU(A(d,m)) in these cases.

A positive definite and integral quadratic form f is called irrecoverable if there is a quadratic form F such that it represents all proper subforms of f, whereas it does not represent f itself. In this case, F is called an isolation of f. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order 4, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results.

In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. These three pieces are hence invariants of the genus, spinor genus, and class, respectively, extending known results for lattices and verifying a conjecture of the first author and Haensch. We furthermore extend the definition of p-neighbors to include lattice cosets and construct an algorithm to compute representatives for the classes in the genus or spinor genus via the p-neighborhoods.

Given a totally real number field F, we show that there are only finitely many totally real extensions of K of a fixed degree that admit a universal quadratic form defined over F. We further obtain several explicit classification results in the case of relative quadratic extensions.

For a (positive definite and integral) quadratic form f, a quadratic form is said to be an isolation of f from its proper subforms if it represents all proper subforms of f, but not f itself. It was proved that the minimal rank of isolations of the square quadratic form x2 is three, and there are exactly 15 ternary diagonal isolations of x2. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares I2=x2+y2, and there are quinary isolations of I2. In this article, we prove that there are at most 231 quinary isolations of I2, which are listed in Table 1. Moreover, we prove that 14 quinary quadratic forms with dagger mark in Table 1 are isolations of I2.

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series.

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.

Lifting problem for universal quadratic forms asks for totally real number fields K$K$ that admit a positive definite quadratic form with coefficients in Z$\mathbb {Z}$ that is universal over the ring of integers of K$K$. In this paper, we show K=Q(zeta 7+zeta 7-1)$K=\mathbb {Q}(\zeta _7+\zeta _7<^>{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.

A (positive definite and integral) quadratic form f is said to be universal if it represents all positive integers, and is said to be primitively universal if it represents all positive integers primitively. We also say f is almost primitively universal if it represents almost all positive integers primitively. Conway and Schneeberger proved (see [1]) that there are exactly 204 equivalence classes of universal quaternary quadratic forms. Recently, Earnest and Gunawardana proved in [4] that among 204 equivalence classes of universal quaternary quadratic forms, there are exactly 152 equivalence classes of almost primitively universal quaternary quadratic forms. In this article, we prove that there are exactly 107 equivalence classes of primitively universal quaternary quadratic forms. We also determine the set of all positive integers that are not primitively represented by each of the remaining 152−107=45 equivalence classes of almost primitively universal quaternary quadratic forms.

Abstract In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy’s lemma. Using this, we show the “pentagonal theorem of 63”, which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 11 , 13 , 14 , 17 , 18 , 19 , 23 , 28 , 31 , 33 , 34 , 39 , 42 , 63 . 1,~{}2,~{}3,~{}4,~{}6,~{}7,~{}8,~{}9,~{}11,~{}13,~{}14,~{}17,~{}18,~{}19,~{}23% ,~{}28,~{}31,~{}33,~{}34,~{}39,~{}42,~{}63. We further show that these integers form a unique minimal universality criterion set.

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes t j Z 2 t_j\mathbb {Z}^2 ) relating the number of lattice points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.

In this article, we consider weighted sums of generalized polygonal numbers with coefficients 1 or 2. We show that for any m≥10, those weighted sums of generalized m-gonal numbers represent every non-negative integers if they only represent 1, m-4, and m-2. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients 1 or 2.

AbstractA (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms $ax^2+by^2+cz^2$ and $ax^2+by^2+cz^2+dw^2$ ’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures. We classify all prime-universal diagonal quadratic forms regardless of rank, and prove the so-called 67-theorem for a diagonal quadratic form to be prime-universal.

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as [Formula: see text] for some integers [Formula: see text]. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when [Formula: see text], which are sums of [Formula: see text] nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer [Formula: see text].

For each positive integer n, let g_Δ(n) be the smallest positive integer g such that every complete quadratic polynomial in n variables which can be represented by a sum of odd squares is represented by a sum of at most g odd squares. In this paper, we analyze g_Δ(n) by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We prove that the growth of g_Δ(n) is at most an exponential of √(n), which is the same as the best known upper bound on the g-invariants of the original quadratic Waring's problem. We also determine the exact value of g_Δ(n) for each positive integer less than or equal to 4.

Let $m, n$ be positive integers with $m\le n$. Let $\kappa(m,n)$ be the largest integer $k$ such that for any (positive definite and integral) quadratic forms $f_1,\ldots,f_k$ of rank $m$, there exists a quadratic form of rank $n$ that represents $f_i$ for any $i$ with $1\le i \le k$. In this article, we determine the number $\kappa(m,n)$ for any integer $m$ with $1\le m\le 8$, except for the cases when $(m,n)=(3,5)$ and $(4,6)$. In the exceptional cases, it will be proved that $1\le \kappa(3,5), \ \kappa(4,6)\le 2$. We also discuss some related topics.

In this article, we study the representability of integers as sums of pentagonal numbers, where a pentagonal number is an integer of the form $P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. In particular, we prove the "pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $11$, $13$, $14$, $17$, $18$, $19$, $23$, $28$, $31$, $33$, $34$, $39$, $42$, and $63$. We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results.

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.

This paper presents the unique challenge and technical solutions for the transportation and installation (T&I) of the fully-integrated SHWE topside. The SHWE topside, one of the heaviest topside in Asia, has a weight of about 26000 MT and is transported to the installation site in the Bay of Bengal, Myanmar by the bottle shaped barge HYSY229. The emphasis of this paper is on describing the most challenging part of the T&I design, the barge strength under topside transport and floatover mating analysis with the long swell condition.

We say a positive integer n satisfies the Lehmer property if phi(n) divides n-1, where phi(n) is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form D-p,D-n = np(n)+1, for a prime p and a positive integer n, or of the form alpha 2(beta) + 1 for alpha <= beta does not satisfy the Lehmer property..