Small prime solutions to cubic Diophantine equations

Author: 
Date created: 
2006
Abstract: 

Let a1, . . . , a9 be non-zero integers and n any integer. Suppose that a1 + ··· + a9 = n (mod 2) and (ai, aj) = 1 for 1 <= i < j <= 9. In this thesis we will prove that (i) if not all of the aj’s are of the same sign, then the cubic diagonal equation a1p1^3 + ··· + a9p9^3 = n has prime solutions satisfying pj << n^{1/3} + max{|aj|}^{20+ e}; and (ii) if all aj are positive and n >> max{|aj|}^{61+e}, then the cubic diagonal equation a1p1^3 + ··· + a9p9^3 = n is soluble in primes pj. This result is motivated from the analogous result for quadratic diagonal equations by S.K.K. Choi and J. Liu. To prove the results we will use the Hardy-Littlewood Circle method, which we will outline. Lastly, we will make a note on some possible generalizations to this particular problem.

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Language: 
English
Document type: 
Thesis
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Copyright remains with the author
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Department: 
Department of Mathematics - Simon Fraser University
Thesis type: 
Thesis (M.Sc.)
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