Computational results on sums of a prime with squares or cubes

Source abstract

One conjectures that all integers greater than 14 are the sum of an odd prime and two positive squares. We prove that all integers $n$ with $14<n<10^{26}$ can be represented in this manner. Along the way, we investigate sums of a prime and one square and then ``bootstrap'' to get our results on sums of a prime and two squares. We also show that all integers between $308$ and $10^{16}$ can be represented as the sum of an odd prime and two positive cubes. Our methods require the determination of the integral points on various elliptic curves.