2010
DOI: 10.1142/s179304211000279x
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Fourteen Octonary Quadratic Forms

Abstract: We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

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Cited by 22 publications
(40 citation statements)
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“…for all partitions i + j + k + l = 8, i, j, k, l ≥ 0. There are a total of 165 such quadratic forms, and out of which 81 quadratic forms (corresponding to i = 0 or l = 0) have already been considered by several authors [3,4,17]. In the second part, we consider the remaining 84 quadratic forms and give formulas for the corresponding representation numbers.…”
Section: Present Workmentioning
confidence: 99%
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“…for all partitions i + j + k + l = 8, i, j, k, l ≥ 0. There are a total of 165 such quadratic forms, and out of which 81 quadratic forms (corresponding to i = 0 or l = 0) have already been considered by several authors [3,4,17]. In the second part, we consider the remaining 84 quadratic forms and give formulas for the corresponding representation numbers.…”
Section: Present Workmentioning
confidence: 99%
“…(i, j, k, l) Type (1,0,1,6), (1,0,3,4), (1,0,5,2), (1,1,1,5), (1,1,3,3), (1,1,5,1), (1,2,1,4), (1,2,3,2), (1,3,1,3), (1,3,3,1), (1,4,1,2), (1,5,1,1), (2,0,0,6), (2,0,2,4), (2,0,4,2), (2,1,0,5), (2,1,2,3), (2,1,4,1), (2,2,0,4), (2,2,2,2), I (2,3,0,3), (2,3,2,1), (2,4,0,2), (2,5,0,1), (3,0,1,4), (3,0,3,2), (3,1,1,3), (3,1,3,3), (3,2,1,2), (3,3,1,1), (4,0,0,4), (4,0,2,2), (4,1,0,3), (4,1,2,1), (4,2,0,2), (4,3,0,1), (5,0,1,2), (5,1,1,1), (6,0,0,2), (6,1,0,1) (1,0,0,7), (1,0,2,5), (1,0,4,3), (1,0,6,1), (1,1,0,6), (1,1,2,4), (1,1,4,2), (1,2,0,5), (1,2,2,3), (1,2,4,1), (1,3,0,4), (1,3,2,2), (1,4,0,3), (1,4,2,1), (1,5,0,2), (1,6,0,1), (2,0,1,5), (2,0,3,3), (2,0,5,1), (2,1,1,4), II (2,1,3,2), (2,2,1,3), (2,2,3,1), (2,3,1,2), (2,4,1,1), (3,0,0,5), (3,0,2,3), (3,0,4,1), (3,1,0,4), (3,1,2,2), (3,2,0,3), (3,2,2,1), (3,3,0,2), (3,4,0,1), (4,0,1,3), (4,0,3,1), (4,1,1,2), (4,2,1,1), (5,0,0,3), (5,0,2,1), (5,1,0,2), (5,2,0,1), (6,0,1,1), (7,0,0,1) Table 2. There are several methods used in the literature to obtain results of this type.…”
Section: Present Workmentioning
confidence: 99%
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“…(2) (x 2 1 + x 1 x 2 + x 2 2 ) + c 1 (x 2 3 + x 3 x 4 + x 2 4 ) + c 2 (x 2 5 + x 5 x 6 + x 2 6 ) + c 3 (x 2 7 + x 7 x 8 + x 2 8 ),…”
Section: Introductionmentioning
confidence: 99%
“…, x 8 ) ∈ Z 8 | n = a 1 x with i + j + k + l = 8 under the conditions i ≡ j ≡ k ≡ l ≡ 0 (mod 2) or i ≡ j ≡ k ≡ l ≡ 1 (mod 2) appeared in literature. See [1], [2], [3], [4], [6], [9] and [11]. For convenience, we write (i, j, k, l) to denote an octonary quadratic form given by (1.1), and we write N(1 i , 2 j , 3 k , 6 l ; n) to denote the number of representations of n by the octonary quadratic form (i, j, k, l).…”
Section: Introductionmentioning
confidence: 99%