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Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers K of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota's unit formula without cohomological results of [C. B. Setzer, Units over totally real C 2 × C 2 fields, J. Number Theory  12 (1980) 160–175]. [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]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

Let gcd (j , k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd -sum function f (gcd (j , k)) and the function ∑ d | k , d s | j (f ∗ μ) (d) for any positive integers j and k , namely ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) and ∑ k ≤ x 1 k s (r + 1) ∑ j = 1 k s j r ∑ d | k d s | j (f ∗ μ) (d) with any fixed integer s > 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd -sum functions f (gcd (j , k)). [ABSTRACT FROM AUTHOR]

For positive integers m , n , k with m ≥ 2 and ⌈ n / 2 ⌉ ≤ k ≤ n , let E { m } (m , n , k) be the sum of multiple zeta values of depth k and weight m n with arguments m or 2 m , i.e. E { m } (m , n , k) = ∑ | α | = n α j = 1 or 2 ζ (m α 1 , m α 2 , ... , m α k). In this paper, we are going to evaluate E { m } (m , n , k). As an application, we produce the stuffle relations from k identical Riemann zeta values ζ (2 m) as well as k identical Riemann zeta values ζ (2 m) and ζ (m). [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]

Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]

A Carmichael number is a composite number such that divides for all integers coprime to . Korselt discovered that is a Carmichael number if and only if is square-free and for each prime divisor of . Let , a -number is defined to be a composite number , such that and for each prime . The set of all such that is a -number is called the Korselt set of and we denote this set by . In this paper, we investigate some properties of when is a power of a prime. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

In this paper, the -Gordian complex of knots is defined. The authors show that for any knot and any positive integer , there is a finite set of knots of size containing , such that the -Gordian distance between any two knots in this set is 1 for all . [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]

Let be an integral domain with quotient field . Call an almost GCD domain (AGCD domain) if for all , there is an integer such that is principal. We say that is a locally AGCD domain (resp., strongly locally AGCD domain) if is an AGCD domain for all maximal ideals of (resp., for all , there is an integer such that is locally principal). In this paper, we study some ring-theoretic properties of locally and strongly locally AGCD domains. Let be an indeterminate over . We use the ring for an integer to give some examples of locally and strongly locally AGCD domains. [ABSTRACT FROM AUTHOR]

In this paper, the convolution sum Σi+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams. [ABSTRACT FROM AUTHOR]

Ramanujan's formula for the number r24(n) of representations of a positive integer n as a sum of 24 squares asserts that where and Ramanujan's function τ(n) is given by In this paper, we determine a class of formulae analogous to Ramanujan's formula for r24(n). [ABSTRACT FROM AUTHOR]