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

(33 reference statements)

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“…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

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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).

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Universal Sums of m-Gonal Numbers

Kane

Liu

2019

International Mathematics Research Notices

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“….). 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, .…”

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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

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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.

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“…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

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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.

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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

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