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Conjecture H of Hardy and Littlewood says that every large integer is either a square or the sum of a prime and a square. Despite progress showing that this is true for almost all large integers, Conjecture H remains unproven. In this paper, Conjecture H is approximated using Piatetski–Shapiro sequences by showing that all large integers can be written in the form p + [ m c ] for any fixed 1 < c < 6 9 / 6 2. The existing results for the conjecture are also extended by showing that almost all large integers can be written in the form [ p c ] + m 2 for any fixed 1 < c < 4 1 9 / 3 5 2. [ABSTRACT FROM AUTHOR]

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 x 2 + y 2 + z 2 for some integers x , y , z ≠ ± 1. 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 k = 3 , which are sums of k nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer k ≥ 3. [ABSTRACT FROM AUTHOR]

Let q be a prime of the form 4 k + 3. Then for any three mutually orthogonal great circles of the sphere x 2 + y 2 + z 2 = q lying on rational planes, at least one of these circles does not contain any rational points. [ABSTRACT FROM AUTHOR]

For positive integers a , b and c , let T (a , b , c ; n) denote the number of representations of a nonnegative integer n as a x 1 (x 1 + 1) / 2 + b x 2 (x 2 + 1) / 2 + c x 3 (x 3 + 1) / 2 where x 1 , x 2 and x 3 are nonnegative integers, and let N (a , b , c ; n) denote the number of representations of n as a x 1 2 + b x 2 2 + c x 3 2 where x 1 , x 2 and x 3 are integers. Recently, Sun proved a number of relations between T (a , b , c ; n) and N (a , b , c ; n) along with numerous conjectures on such relations. In this work, we confirm several conjectures of Sun by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including 1). For example, we show that each n = 1 , 2 , 3 , ... can be written as x 2 + y 2 + z 2 + w 2 (x , y , z , w ∈ ℕ = { 0 , 1 , 2 , ... }) with | x + y − z | ∈ { 4 k : k ∈ ℕ } (or | 2 x − y | ∈ { 4 k : k ∈ ℕ } , or x + y − z ∈ { ± 8 k : k ∈ ℕ } ∪ { 0 } ⊆ { t 3 : t ∈ ℤ }), and that we can write any positive integer as x 2 + y 2 + z 2 + w 2   (x , y , z , w ∈ ℤ) with x + y + 2 z (or x + 2 y + 2 z) a power of four. We also prove that any n ∈ ℕ can be written as x 2 + y 2 + z 2 + 2 w 2 (x , y , z , w ∈ ℤ) with x + y + z + w a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer n > 1 can be written as a 2 + b 2 + 3 c + 5 d with a , b , c , d ∈ ℕ. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cyclicity of the 2 -class group of the first Hilbert 2 -class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let p 1 ≡ p 2 ≡ − q ≡ 1 (mod 4) be different prime integers. Put 𝕜 = ℚ ( p 1 p 2 q) , and denote by C 𝕜 , 2 its 2 -class group and by 𝕜 2 (1) (respectively 𝕜 2 (2) ) its first (respectively second) Hilbert 2 -class field. Then, we are interested in studying the metacyclicity of G = Gal (𝕜 2 (2) / 𝕜) and the cyclicity of Gal (𝕜 2 (2) / 𝕜 2 (1)) whenever the 4 -rank of C 𝕜 , 2 is 1. [ABSTRACT FROM AUTHOR]

Let R (n) denote the number of representations of an even integer n , n ≢ 2 (mod 3) as the sum of two squares, one cube and three fourth powers of primes, and by ℰ (N) we denote the number of even integers n ≤ N , n ≢ 2 (mod 3) such that the expected asymptotic formula for R (n) fails to hold. In this paper, it is proved that ℰ (N) ≪ N 6 1 1 4 4 + 𝜀 for any 𝜀 > 0. [ABSTRACT FROM AUTHOR]

In a recent paper, Sun posed six conjectures on the relations between T (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; n) , where T (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 (x 1 + 1) 2 + a 2 x 2 (x 2 + 1) 2 + ⋯ + a k x k (x k + 1) 2 , where a 1 , a 2 , ... , a k are positive integers, n , x 1 , x 2 , ... , x k are arbitrary nonnegative integers, and N (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , where this time x 1 , x 2 , ... , x k are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

Over the ring of algebraic integers 𝒪 of a number field K , the quadratic congruence a 1 x 1 2 + ⋯ + a k x k 2 ≡ c modulo a nonzero ideal ℐ is considered. We prove explicit formulas for N k (a 1 , ... , a k , c , ℐ) and N k ∗ (a 1 , ... , a k , c , ℐ) , the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k and the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k with x 1 x 2 ⋯ x k coprime to ℐ , respectively. If ℐ is contained in a prime ideal 𝒫 containing the rational prime p , it is assumed that 𝒫 is unramified over p. Moreover, some interesting identities for exponential sums are proved. [ABSTRACT FROM AUTHOR]

We determine primitive solutions to the equation (x − r) 2 + x 2 + (x + r) 2 = y n for 1 ≤ r ≤ 1 0 4 , making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier. [ABSTRACT FROM AUTHOR]

Let p¯(n) denote the number of overpartitions of n. Recently, several infinite families of congruences modulo 12, 27 and 40 for p¯(n) have been established. In this paper, we investigate systematically some other infinite families of congruences modulo 12, 24 and 40 for p¯(n). [ABSTRACT FROM AUTHOR]

For each integer m ≥ 3 , let P m (x) denote the generalized m -gonal number (m − 2) x 2 − (m − 4) x 2 with x ∈ ℤ. Given positive integers a , b , c , k and an odd prime number p with p ∤ c , we employ the theory of ternary quadratic forms to determine completely when the mixed sum a x 2 + b y 2 + c P p k + 2 (z) represents all but finitely many positive integers. [ABSTRACT FROM AUTHOR]

Let be an integer greater than or equal to . We show that if a multiplicative function satisfies for all positive integers , then is the identity function. [ABSTRACT FROM AUTHOR]

Let denote the number of partition triples of where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for . Moreover, let denote the number of representations of a non-negative integer in the form with We find three arithmetic relations between and , such as . [ABSTRACT FROM AUTHOR]

The theory of quasimodular forms is used to evaluate the convolution sums ... σ(l) σ(m), ... σ(l) σ(m) and ... σ(l) σ(m) for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form x12 + x22 + x32 + x42 + 5(x52 + x62 + x72 + x82 is determined. [ABSTRACT FROM AUTHOR]

We find a basis for the space Sk(Γ1(4)) of cusp forms of weight k for the congruence subgroup Γ1(4) in terms of Eisenstein series. As an application, we obtain formulas for r2k(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares. [ABSTRACT FROM AUTHOR]

Using known formulas for R(a,b)(.n). the number of representations of n as a times a square plus b times a square, we prove 21 conjectured formulas of Melham on the number of representations of n as sums of triangular, pentagonal and heptagonal numbers. We also demonstrate how our technique can be used to prove the other 277 conjectured formulas of Melham concerning representations by other figurate numbers. [ABSTRACT FROM AUTHOR]

We consider the sum of squared absolute values of the Dirichlet L-function taken at the nontrivial zeros of another Dirichlet L-function. [ABSTRACT FROM AUTHOR]

In this paper, using the (p, fc)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit formulas are obtained for the number of representations of a positive integer n by the quadratic forms ... with a ≠ 0, a + b + c = 4 and ... with k + I = 2 and r + s + i = 2 by employing these identities. [ABSTRACT FROM AUTHOR]

Let rs(n) denote the number of ways to write an integer n as the sum of s squares of integers. In this paper, the congruent properties for rs(n) are studied. We give the elementary combinatorial proofs of all related results due to Wagstaff and Chen, and obtain some new results. [ABSTRACT FROM AUTHOR]

Let rk(n) and tk(n) denote the number of representations of an integer n as a sum of k squares, and as a sum of k triangular numbers, respectively. We prove that $$ t_8(n) = \frac{1}{2^{10}\times 3^{2}} (r_8(8n+8) - 16r_8(2n+2)), $$ and therefore the study of the sequence t8(n) is reduced to the study of subsequences of r8(n). We give an additional 21 analogous results for sums of squares and sums of triangular numbers induced by partitions of 8. We give a brief indication of what happens for the case k ≥ 9. [ABSTRACT FROM AUTHOR]

Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial. [ABSTRACT FROM AUTHOR]