Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Colloquium Mathematicum

2016 | 145 | 1 | 149-155

## On some universal sums of generalized polygonal numbers

EN

### Abstrakty

EN
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = {0,1,2,...} has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as $p₃(x) + p₅(y) + p_{11}(z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that $p₃ + p₅ + p_{11}$ and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.

149-155

wydano
2016

### Twórcy

autor
• Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A.
autor
• Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China