Universal sums of generalized pentagonal numbers

Ju, Jangwon

doi:10.1007/s11139-019-00142-3

Cited by 21 publications

(25 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When m = 4, we can deduce that γ 4 = 15 from the Conway-Schneeberger Fifteen Theorem and the fact that the quaternary quadratic form x 2 + 2x 2 + 5x 2 + 5x 2 represents every positive integer except for 15. For m = 5, Ju [14] recently showed that γ 5 = 109. Since each hexagonal number can be written as a It may be natural to ask whether one can consider similar questions with mixed sums of m j -gonal numbers.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Universal Sums of m-Gonal Numbers

Kane

Liu

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of m) on the size of the set Sm ⊂ N such that if a sum of m-gonal numbers represents Sm, then it represents N. 1 triangular number and vice versa, we conclude for the m = 6 case that γ 6 = γ 3 = 8. The m = 8 case has been resolved by Ju and Oh [15], who proved that γ 8 = 60. Having established a number of individual cases, it is then natural to ask about the growth of γ m as a function of m. Theorem 1.1. (1) For m ≥ 3 and every ε > 0, there exists an absolute (effective) constant C ε such that γ m ≤ C ε m 7+ε . (2) There is no uniform upper bound which holds for all m. Specifically, if m ≥ 6, then γ m ≥ m − 4 and for every element ℓ ∈ N there exists a sum of generalized polygonal numbers which represents every nonnegative integer except for ℓ.Remark. Theorem 1.1 (2) is due to Guy [12] and is by explicit construction for the form with a j = 1, but we include it here for comparison with the upper bound obtained in Theorem 1.1 (1).

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Universal Sums of m-Gonal Numbers

Kane

Liu

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“….). Let (b, c, d) be one of the triples (1, 1, 2), (1, 2, 3), (1, 2, 6) and (2,3,4). We show that each n = 0, 1, 2, .…”

mentioning

confidence: 90%

“…contains all nonnegative integers whenever it contains the twelve numbers 1,3,8,9,11,18,19,25,27,43, 98, 109.…”

Section: Introductionmentioning

confidence: 99%

On sums of four pentagonal numbers with coefficients

Krachun

Sun

2020

era

View full text Add to dashboard Cite

The pentagonal numbers are the integers given by p 5 (n) = n(3n − 1)/2 (n = 0, 1, 2, . . .). Let (b, c, d) be one of the triples (1, 1, 2), (1, 2, 3), (1, 2, 6) and (2,3,4). We show that each n = 0, 1, 2, . . . can be written as w+bx+cy+dz with w, x, y, z pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.

show abstract

“…A tight universal m-gonal form having minimum 1 is simply called universal. The universal m-gonal forms have been studied by many mathematicians and there are several results on the classification problem (see, for example, [2], [7], [8] and [9]). Note that P 4 pxq " x 2 and the classification of universal diagonal quadratic forms can be easily done by using Conway-Schneeberger 15-Theorem(see [1] and [3]).…”

Section: Introductionmentioning

confidence: 99%

Tight universal triangular forms

Kim¹

2021

Preprint

View full text Add to dashboard Cite

For a subset S of nonnegative integers and a vector a " pa 1 , . . . , a k q of positive integers, let V 1 S paq " ta 1 s 1 `¨¨¨`a k s k : s i P Suzt0u. For a positive integer n, let T pnq be the set of integers greater than or equal to n. In this paper, we consider the problem of finding all vectors a satisfying V 1 S paq " T pnq, when S is the set of (generalized) m-gonal numbers and n is a positive integer. In particular, we completely resolve the case when S is the set of triangular numbers.

show abstract

Universal sums of generalized pentagonal numbers

Abstract: For an integer x, an integer of the form P 5 pxq " 3x 2´x

Cited by 21 publications

References 13 publications

Universal Sums of m-Gonal Numbers

Universal Sums of m-Gonal Numbers

On sums of four pentagonal numbers with coefficients

Tight universal triangular forms

Contact Info

Product

Resources

About