On perfect powers that are sums of cubes of a seven term arithmetic progression
نویسندگان
چکیده
منابع مشابه
Perfect Powers That Are Sums of Consecutive Cubes
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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We show that every integer between 1290741 and 3.375× 1012 is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes.
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Let n ≥ 3. This paper is concerned with the equation a3 + b3 = cn, which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer–Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density 28219 44928 ≈ 0.628, a...
متن کاملOn 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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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2020
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2020.04.020