A quantitative Erdös-Fuchs theorem and its generalization
A generalization of the classical circle problem
A basis of ℤₘ, II
Given a set A ⊂ ℕ let denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and for all n̅ ∈ ℤₘ.
The Diophantine equation
Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation has only the solution (x,y,z) = (1,1,1). We give an extension of this result.
On near-perfect numbers
For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².
A basis of Zₘ
Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and for all n̅ ∈ Zₘ.
On near-perfect and deficient-perfect numbers
For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.
Some results on Poincaré sets
It is known that a set of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if , where and denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set by replacing with . It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.
Note on a result of Haddad and Helou.
Unique difference bases of .
Idempotence-preserving maps between matrix spaces over fields of characteristic 2.
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