In this paper, we shall use some simple inequalities and a deep result on the existence of primitive divisors of Lucas numbers to prove that the exponential Diophantine equation x y + y z = z x has no positive integer solution (x, y, z) with 2 | y. MSC: 11D61

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The increasing need to reduce damage and downtime in modern buildings has led to the development of a low-damage design philosophy, where the earthquake loads can be resisted with damage confined to easily replaceable components. Post-tensioned (PT) concrete walls have emerged as a popular lowdamage structural system that have been implemented in a range of buildings. In order to provide essential evidence to support the development of lowdamage concrete structures, a system-level shake-table test was conducted on a two-storey low-damage concrete wall building implementing state-of-art design concepts. The test building included PT rocking walls that provide the primary lateral-load resistance in both directions, a frame that utilized slotted beam connections, and a range of alternative energy dissipation devices that were installed at wall base or/and beam-column joints. The building was subjected to 39 tests with a range of intensity ground motions, incorporating both unidirectional and bidirectional ground motions on the structure with different combinations of wall strength and energy dissipating devices. The building performed exceptionally well during the intense series of tests, confirming the suitability of both the design methods and the connection detailing implemented. The building achieved an immediate occupancy performance objective even when subjected to maximum considered earthquake hazard shaking. The building exhibited only minor damage at the conclusion of testing, with distributed cracking in the floors and cosmetic spalling in the wall toes that did not compromise structural capacity or integrity and could be easily repaired with minimal disruption.The test has provided a rich dataset that is available for further analysis of the building response and validation of design methods and numerical models.

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