I'm trying to solve the exercise 4 of chapter 2 of vaughan's book, i want to show that every large positive integer is the sum of one square and seven cubes. Can somebody give me the solution orm at leastm an hint? thank you in advance
2026-04-02 03:42:48.1775101368
one square and seven cubes, circle method
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Unfortunatly i can't post comments, so this has to do. I think you should provide more information about your atempts to solve the problem. That is give the definition of your major arcs. What is the expected asymptotic for the representation of an integer N as sum of 7 cubes and one square? As for the minor arcs - we just have to save something over the major arcs contribution.
My first guess would be that the minor arcs can be dealt with an application of Hölder and Hua's estimate for the disentangled minor arcs integral. The singular series should also not be a severe problem as we have rather many variables.
As mentioned earlier you should just give more details about what you tried; i can edit some thoughts in later.
leithian