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Representation functions with different weights

100%

Yang Q.-H.

Colloquium Mathematicum

2014

tom 137

nr 1

1-6

For any given positive integer k, and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ and $r_{1,l}(A,n) = r_{1,l}(ℕ ∖ A,n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

The Diophantine equation $(bn)^{x} + (2n)^{y} = ((b+2)n)^{z}$

63%

Tang M., Yang Q.-H.

Colloquium Mathematicum

2013

tom 132

nr 1

95-100

Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^{x} + 2^{y} = (b+2)^{z}$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Yang Q.-H.

1 Tang M.

1 2014

1 2013

Wyniki wyszukiwania

Representation functions with different weights

The Diophantine equation $(bn)^{x} + (2n)^{y} = ((b+2)n)^{z}$