On Waring’s problem: Three cubes and a sixth power
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We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.
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2004 ◽
Vol 76
(3)
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pp. 303-316
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Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences
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1993 ◽
Vol 345
(1676)
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pp. 385-396
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1933 ◽
Vol 232
(707-720)
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pp. 1-26
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2009 ◽
Vol 51
(3)
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pp. 703-712
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1991 ◽
Vol 109
(2)
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pp. 229-256
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1969 ◽
Vol 65
(2)
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pp. 445-446
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Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences
◽
1993 ◽
Vol 345
(1676)
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pp. 327-338
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