On the Diophantine equation (x-1)^k+x^k+(x+1)^k=y^n

Zhang, Zhongfeng

doi:10.5486/pmd.2014.5815

Cited by 118 publications

(159 citation statements)

References 9 publications

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“…11 This genus was first characterized in 2014 following reclassification of members of the genus Thalassomonas. 11 There have been only 10 species of Thalassotalea discussed in the literature and they have been primarily studied for their agarolytic properties, commensalism with marine animals, and pathogenic contribution to coral diseases. 11–16 To date, there have been no published secondary metabolite studies on this genus.…”

mentioning

confidence: 99%

N-Acyl Dehydrotyrosines, Tyrosinase Inhibitors from the Marine Bacterium Thalassotalea sp. PP2-459

et al. 2016

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Thalassotalic acids A–C and thalassotalamides A and B are new N-acyl dehydrotyrosine derivatives produced by Thalassotalea sp. PP2–459, a Gram-negative bacterium isolated from a marine bivalve aquaculture facility. The structures were elucidated via a combination of spectroscopic analyses emphasizing two-dimensional NMR and high-resolution mass spectral data. Thalassotalic acid A (1) displays in vitro inhibition of the enzyme tyrosinase with an IC50 value (130 μM) that compares favorably to the commercially-used control compounds kojic acid (46 μM) and arbutin (100 μM). These are the first natural products reported from a bacterium belonging to the genus Thalassotalea.

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confidence: 99%

N-Acyl Dehydrotyrosines, Tyrosinase Inhibitors from the Marine Bacterium Thalassotalea sp. PP2-459

et al. 2016

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“…Zhongfeng Zhang [21] showed that the only perfect powers that are sums of three consecutive cubes are precisely those already noted by Euler (1), Lucas (2) and Cassels (3). Zhang's approach is write the problem as (4) y n = (x − 1) 3 + x 3 + (x + 1) 3 = 3x(x 2 + 2), and apply a descent argument that reduces this to certain ternary equations that have already been solved in the literature.…”

Section: Introductionmentioning

confidence: 71%

“…By the conclusion of Lemma 4.1, we know that y 1 , y 2 are integers. It follows from (21) that S | y 1 where…”

Section: Frey-hellegouarch Curve For the Case R = Tmentioning

confidence: 99%

Perfect Powers That Are Sums of Consecutive Cubes

2016

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Abstract. Euler noted the relation 6 3 = 3 3 + 4 3 + 5 3 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.

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“…They first proved that all integer solutions of equation (1.4) such that n > 1 and d = x are (x, y) = (0, 0), (x, y, n) = (1, ±2, 2), (2, ±5, 2), (24, ±182, 2) or (x, y) = (−1, −1) with 2 ∤ n. Secondly, they showed that if p ≡ ±5 (mod 12) is prime, p | d and v p (d) ≡ 0 (mod n), then equation (1.4) has no integer solution (x, y) with k = 2. In 2014, the equation (x − 1) k + x k + (x + 1) k = y n x, y, n ∈ Z, n ≥ 2, (1.5) was solved completely by Zhang [22], for k = 2, 3, 4 and the next year, Bennett, Patel and Siksek [4], extend Zhang's result, completely solving equation (1.5) in the cases k = 5 and k = 6. In 2016, Bennett, Patel and Siksek [5], considered the equation (1.4).…”

Section: Introductionmentioning

confidence: 99%