On some universal sums of generalized polygonal numbers

• Autorzy: Fan Ge , Zhi-Wei Sun
• Języki publikacji: 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.

Kategorie tematyczne

• 11B75: Other combinatorial number theory
• 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
• 11P32: Goldbach-type theorems; other additive questions involving primes
• 11D85: Representation problems
• 11E25: Sums of squares and representations by other particular quadratic forms

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2016
• Tom: 145
• Numer: 1
• Strony: 149-155

Daty

wydano

2016

Twórcy

autor

Fan Ge

• Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A.

autor

Zhi-Wei Sun

• Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China

Identyfikatory

DOI

10.4064/cm6742-3-2016