Regular positive ternary quadratic forms

Oh, Byeong-Kweon

doi:10.4064/aa147-3-3

Cited by 41 publications

(66 citation statements)

References 10 publications

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“…All but 22 of the 913 candidates were verified to be regular in [7]. Recently the author proved in [10] Therefore we have…”

Section: Corollary 22 Under the Same Notations As Above Assume Thamentioning

confidence: 97%

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Ternary Universal Sums of Generalized Pentagonal Numbers

Oh¹

2011

Journal of the Korean Mathematical Society

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with x ∈ Z is said to be a generalized m-gonal number. Let a ≤ b ≤ c and k be positive integers. The quadruple (k, a, b, c) is said to be universal if for every nonnegative integer n there exist integers x, y, z such that n = ap k (x)+bp k (y)+cp k (z). Sun proved in [16] that, when k = 5 or k ≥ 7, there are only 20 candidates for universal quadruples, which he listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He verified that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only (5, 1, 1, t) for t = 6, 8, 9, 10, (5, 1, 2, 8) and (5, 1, 3, s) for s = 7, 8 as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.

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“…All but 22 of the 913 candidates were verified to be regular in [7]. Recently the author proved in [10] Therefore we have…”

Section: Corollary 22 Under the Same Notations As Above Assume Thamentioning

confidence: 97%

“…Under this situation we use the method on the representations of ternary forms which is developed in [10]. It seems to be quite a difficult problem to determine whether or not p…”

Section: Introductionmentioning

confidence: 99%

Ternary Universal Sums of Generalized Pentagonal Numbers

Oh¹

2011

Journal of the Korean Mathematical Society

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“…As a natural generalization of Ramanujan's result on diagonal quaternary quadratic forms representing all nonnegative integers, it was proved in [5] and [13] that there are exactly 40 quaternary sums of generalized octagonal numbers which represent all nonnegative integers. Furthermore, the "octagonal theorem of sixty" was proved in [5], which states that a sum a 1 P 8 px 1 q`a 2 P 8 px 2 q`¨¨¨`a k P 8 px k q of generalized octagonal numbers represents all nonnegative integers if and only if it represents 1,2,3,4,6,7,9,12,13,14,18, and 60, which might be considered this as a natural generalization of the "15-theorem". In this article, we generalize the above theorem to the mixed sum of generalized 4-and 8-gonal numbers.…”

Section: Introductionmentioning

confidence: 99%

“…Here, a universal sum Φ of generalized 4-and 8-gonal numbers is called proper if there does not exist a proper sum of Φ that is universal. Furthermore, in Section 5, we prove that a mixed sum of generalized 4and 8-gonal numbers is universal if and only if it represents the integers 1,2,3,4,5,6,7,8,9,10,12,13,14,15,18,20,30,60, and 61, which might also be considered as a generalization of the "15-theorem". Let f px 1 , x 2 , .…”

Section: Introductionmentioning

confidence: 99%

Universal mixed sums of generalized 4- and 8-gonal numbers

2019

Int. J. Number Theory

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An integer of the form Pmpxq " pm´2qx 2´p m´4qx 2 for an integer x, is called a generalized m-gonal number. For positive integers α 1 , . . . , αu and β 1 , . . . , βv, a mixed sum Φ " α 1 P 4 px 1 q`¨¨¨`αuP 4 pxuq`β 1 P 8 py 1 q`¨¨¨β v P 8 pyv q of generalized 4-and 8-gonal numbers is called universal if Φ " N has an integer solution for any nonnegative integer N . In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized 4-and 8-gonal numbers. Furthermore, the "61-theorem" is proved, which states that an arbitrary mixed sum of generalized 4-and 8-gonal numbers is universal if and only if it represents the integers

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“…It was Dickson [5] who first studied regular quadratic forms systematically, and later Watson [17,18] showed that there are only finitely many similarity classes of positive definite regular ternary quadratic forms. Enumeration of these similarity classes has been undertaken by several authors [10,12,13]. The enumeration is almost completed, although the regularity of several candidates has not to date been proven without assuming a form of the Generalized Riemann Hypothesis [12].…”

Section: Introductionmentioning

confidence: 99%