Sums and Differences of Three k-th Powers
| dc.creator | Heath-Brown, D. R. | |
| dc.date | 2008-06-26 | |
| dc.date.accessioned | 2026-07-07T09:46:56Z | |
| dc.date.available | 2026-07-07T09:46:56Z | |
| dc.description | Let k>2 be a fixed integer exponent and let θ> 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^θN^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take θstrictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley. | |
| dc.identifier | https://arxiv.org/abs/0806.4330 | |
| dc.identifier | http://arxiv.org/abs/0806.4330 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163699 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11D45 | |
| dc.title | Sums and Differences of Three k-th Powers | |
| dc.type | text |