2013
DOI: 10.4064/cm132-1-7
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The Diophantine equation (bn)x+(2n)y=((b+2)n)z

Abstract: THE DIOPHANTINE EQUATION (bn) x + (2n) y = ((b + 2)n) z BY MIN TANG (Wuhu) and QUAN-HUI YANG (Nanjing) Abstract. Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b = 89, the Diophantine equation b x + 2 y = (b + 2) z has only the solution (x, y, z) = (1, 1, 1). We give an extension of this result.

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Cited by 12 publications
(18 citation statements)
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“…[ 123 ] By doing just that, layer-by-layer assembly has been able to realize composites with incredible strength and gas barrier. [ 51,113,114 ] LbL has the unique Reproduced with permission. [ 9 ] b) TEM image of a cross-section of a polymer/clay LbL fi lm deposited onto PET, where dark horizontal lines are clay platelets.…”
Section: Nanoclays In Layer-by-layer Assemblymentioning
confidence: 99%
See 1 more Smart Citation
“…[ 123 ] By doing just that, layer-by-layer assembly has been able to realize composites with incredible strength and gas barrier. [ 51,113,114 ] LbL has the unique Reproduced with permission. [ 9 ] b) TEM image of a cross-section of a polymer/clay LbL fi lm deposited onto PET, where dark horizontal lines are clay platelets.…”
Section: Nanoclays In Layer-by-layer Assemblymentioning
confidence: 99%
“…The fi rst known report of LbL was the alternate adsorption of micrometer-scale colloidal particles of opposite charges. [ 55 ] Nanoparticle-based LbL fi lms have since been shown to exhibit electrical conductivity, [95][96][97][98][99][100] super hydrophobicity, [101][102][103][104][105] water repellency, [ 106 ] UV absorption, [ 107,108 ] optical fi ltering, [ 109 ] fl ame resistance, [ 110,111 ] strength, [112][113][114][115] and gas barrier [ 50,57,[116][117][118][119][120][121][122] by way of carbon nanotubes, spherical nanoparticles (e.g., TiO 2 and SiO 2 ), or two-dimensional nano platelets (e.g., clay and graphene).…”
Section: Introduction To Layer-by-layer Assemblymentioning
confidence: 99%
“…[9][10][11] The selective transport of these ions is of central importance for muscle contraction and expansion. Up to date, Nafion [12,13] and sulfonated polymers such as poly(arylene ether sulfone)s, [14] poly(styrene)s, [15] poly(phenyl sulfone)s, [16] and a sulfonated poly(ether ether ketone)-poly(vinylidene fluoride) [17] blend have been reported in the literature as IPMC actuator materials. However, so far ion selective actuation, as natural muscles do, has not been reported in the literature.…”
Section: Introductionmentioning
confidence: 99%
“…Another parameter for controlling the calcifi cation is the pH value. During the calcifi cation process, a pH change in alkaline direction is monitored in the polymer fi lm as well as in the surrounding solution, caused by the cleavage of urea to CO 3 2− and NH 4 + (Figure 1 ). The pH value of the surrounding solution reaches values of up to 9, after a few minutes of calcifi cation.…”
Section: Resultsmentioning
confidence: 99%
“…The layer-by-layer technique allows the production of very regular polymer fi lms in microscopic dimensions. [ 3 ] These fi lms afford CaCO 3 -coatings on their surface using the gas-diffusion method, in which CO 2 diffuses into a Ca 2+ -rich solution. [ 4 ] An interesting approach in this fi eld a macroscopic hydrogel throughout the whole material.…”
Section: Introductionmentioning
confidence: 99%