Mathematics – Number Theory
Scientific paper
2008-06-26
Mathematics
Number Theory
Scientific paper
Let k>2 be a fixed integer exponent and let \theta > 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^{\theta}N^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take \theta strictly 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.
Heath-Brown D. R.
No associations
LandOfFree
Sums and Differences of Three k-th Powers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Sums and Differences of Three k-th Powers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sums and Differences of Three k-th Powers will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-246598