International Journal of Number Theory

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 𝔅_k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 𝔅₅(n), 𝔅₇(n) and 𝔅₁₁(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 𝔅₅(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 𝔅₉(n) by establishing the generating functions of 𝔅₉(An+B) modulo 4 for some values of A and B.

In this work, we establish formulas for the representations of binary quadratic forms of discriminants -44, -92, -108, -135 and -140 as a sum of Lambert series and eta-quotients.

We consider a family of dynamically defined subsets of Rauzy fractals in the plane. These sets were introduced in the context of the study of symmetries of Rauzy fractals. We prove that their Hausdorff dimensions form an ultimately increasing sequence of numbers converging to 2. These results answer a question stated by the third author in 2012.

For each positive integers n, let g(n) be the number of arithmetic expressions evaluating to n and involving only the constant 1, additions and multiplications, with the restriction that multiplication by 1 is not allowed. We consider two arithmetic expressions to be equal if one can be obtained from the other through a repeated application of the commutative and associative properties. We give an algorithm to compute g(n) and prove that , as n → +∞, where β ≔ log(24)/24.

Let D₁, D₂, D, k, λ be fixed integers such that D₁ ≥ 1, D₂ ≥ 1, gcd(D₁, D₂) = 1, D = D₁D₂ is not a square, ∣k∣ > 1, gcd(D, k) = 1 and λ = 1 or 4 according as 2 ∤ k or not. In this paper, we prove that every solution class S(l) of the equation D₁x²-D₂y² = λk^z, gcd(x, y) = 1, z > 0, has a unique positive integer solution satisfying and , where z runs over all integer solutions (x,y,z) of S(l),(u₁,v₁) is the fundamental solution of Pell's equation u² - Dv² = 1. This result corrects and improves some previous results given by M. H. Le.

We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

Let 𝒜 = {a₁ < a₂ < a₃ < ⋯ < a_n < ⋯} be an infinite sequence of nonnegative integers and let R₂(n) = |{(i, j) : a_i + a_j = n; a_i, a_j ∈ 𝒜; i ≤ j}|. We define . We prove that if the L^∞-norm of is small, then the L¹-norm of is large.

Let p₁ ≡ p₂≡ -q ≡ 1 (mod 4) be different primes such that . Put d = p₁p₂q and , then the bicyclic biquadratic field has an elementary abelian 2-class group, Cl₂(𝕜), of rank 3. In this paper, we study the principalization of the 2-classes of 𝕜 in its 14 unramified abelian extensions 𝕂_j and 𝕃_j within , that is the Hilbert 2-class field of 𝕜. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group of the second Hilbert 2-class field of 𝕂. Additionally, the abelian type invariants of the groups Cl₂(𝕂_j) and Cl₂(𝕃_j) and the length of the 2-class tower of 𝕜 are given.

We investigate a relationship between nondegeneracy of a simple abelian variety A over an algebraic closure of ℚ and of its reduction A₀. We prove that under some assumptions, nondegeneracy of A implies nondegeneracy of A₀.

We establish an inequality comparing the height and the χ-arithmetic volume of toric metrized divisors on . This gives a partial answer to a question of Burgos, Moriwaki, Philippon and Sombra.

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p^∞-Selmer group of E over any strongly admissible p-adic Lie extension K_∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K_∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence is given by the recurrence u_n = 2u_n-1 + u_n-2 with initial condition u₀ = 0, u₁ = 1 and its associated Pell–Lucas sequence is given by the recurrence v_n = 2v_n-1 + v_n-2 with initial condition v₀ = 2, v₁ = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd(n, d) = 1. We prove that the only solutions of the Diophantine equation u_nu_n+d⋯u_n+(k-1)d = y^m are given by u₇ = 13² and u₁u₇ = 13² and the equation v_nv_n+d⋯v_n+(k-1)d = y^m has no solution. In fact, we prove a more general result.

The Calkin–Wilf tree is a rooted infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form a/b, where a and b are relatively prime positive integers. For every 2 × 2 matrix with nonnegative integral coordinates and nonzero determinant, it is possible to construct an analogous tree with this root. If the root is the identity matrix, then the tree consists of all matrices with determinant 1, and this tree possesses the basic properties of the Calkin–Wilf tree of positive rational numbers. The set of all matrices with nonnegative integral coordinates and nonzero determinant decomposes into a forest of rooted infinite binary trees.

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c² + d² with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n²p = x² + qy² with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)^[p/8] (mod p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)^[p/8] (mod p).

Klein's method of solving algebraic equations is discussed and generalized to provide conditions for the unnecessity of the so-called accessory irrationality. We use the modular curve of level N to produce a "hyper-radical" of level N and discuss the accessory irrationalities involved in solving polynomial equations by means of the algebraic hypergeometric functions that define this hyper-radical. The quintic normal forms of Brioschi and Hermite elegantly fit into this framework, and we find explicit conditions for the unnecessary of accessory irrationalities for these normal forms.

It is well known that for any prime p ≡ 3 (mod 4), the class numbers of the quadratic fields and and h(-p), respectively, are odd. It is natural to ask whether there is a formula for h(p)/h(-p) modulo powers of 2. We show the formula h(p) ≡ h(-p)m(p) (mod 16), where m(p) is an integer defined using the "negative" continued fraction expansion of . Our result solves a conjecture of Richard Guy.