Integers that are sums of two rational sixth powers
نویسندگان
چکیده
Abstract We prove that $164\, 634\, 913$ is the smallest positive integer a sum of two rational sixth powers, but not powers. If $C_{k}$ curve $x^{6} + y^{6} = k$ , we use existence morphisms from to elliptic curves, together with Mordell–Weil sieve, rule out points on for various k .
منابع مشابه
On powers that are sums of consecutive like powers
1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example...
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Euler noted the relation 63= 33+43+53 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 determi...
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ژورنال
عنوان ژورنال: Canadian mathematical bulletin
سال: 2022
ISSN: ['1496-4287', '0008-4395']
DOI: https://doi.org/10.4153/s0008439522000157